# Pierre Hansen

Back## Publications

### Cahiers du GERAD

Spectral properties of threshold graphs

**Pierre Hansen**

In this paper we study the spectral properties of the threshold graphs. In particular, we give lower and upper bounds for the largest and smallest eigenvalue...

BibTeX reference**Pierre Hansen**

The energy of a graph `\(G\)`

, denoted by `\({\cal E}(G)\)`

, is defined as the sum of the absolute values of all eigenvalues of `\(G\)`

. In this paper we stu...

In the present paper, we prove lower and upper bounds for each of the ratios `\(GA/\delta\)`

, as well as a lower bound on `\(GA/\sqrt{\delta}\)`

, in terms of...

A small polygon is a polygon of unit diameter.
The question of finding the largest area of small `\(n-\)`

gons
has been answered for some values of `\(n\)`

....

Distributed integral column generation

The Integral Simplex Using Decomposition (ISUD) algorithm has been developed recently to solve large set partitioning problems (SPPs) in a primal way, i.e.,...

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Clustering is the subject of active research in several fields such as operations research, statistics, pattern recognition, and machine learning. The range ...

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Clustering is an automated and powerful technique for data analysis. It aims to divide a given set of data points into clusters which are homogeneous and/o...

BibTeX reference**Pierre Hansen**

Let `\({\mathcal D(G)}\)`

, `\({\mathcal D}^L(G)={\mathcal Diag(Tr)} - {\mathcal D(G)}\)`

and `\({\mathcal D}^Q(G)={\mathcal Diag(Tr)} + {\mathcal D(G)}\)`

b...

**Pierre Hansen**

Let `\(G\)`

be a graph of order `\(n\)`

. The energy `\(\mathcal{E}(G)\)`

of a simple graph `\(G\)`

is the sum of
absolute values of the eigenvalues of its ...

Given an integer solution, the integral simplex using decomposition (ISUD) seeks a descent direction that leads to an improved adjacent integer solution. It ...

BibTeX reference**Pierre Hansen**, Alain Hertz, Cherif Sellal, Damir Vukicevic, Mustapha Aouchiche, and Gilles Caporossi

Necessary and sufficient conditions are provided for the existence of a simple
graph, or a simple connected graph with given
numbers `\(m_{ij}\)`

of edges ...

**Pierre Hansen**

The distance, distance Laplacian and distance signless Laplacian spectra of a connected graph `\(G\)`

are the spectra of the distance, distance Laplacian and...

On the nullity number of graphs

**Pierre Hansen**

The paper discusses bounds on the nullity number of graphs. It is proved in [B. Cheng and B. Liu, *On the nullity of graphs*. Electron. J. Linear Algebra 16 ...

Graph theoretical heuristics are used extensively in many fields to solve approximately large scale optimization problems. Graph theoretical heuristics can a...

BibTeX reference**Pierre Hansen**

In the present paper we are interested in the study of the distance Laplacian eigenvalues of a connected graph with fixed order `\(n\)`

and chromatic number ...

**Pierre Hansen**

The geometric-arithmetic index `\(GA\)`

of a graph `\(G\)`

is the sum of ratios, over all edges of `\(G\)`

, of the geometric mean to the arithmetic mean of t...

**Pierre Hansen**

In the present paper, we prove lower and upper bounds for each of the ratios `\(GA/\delta\)`

, `\(GA/\overline{d}\)`

and `\(\Delta\)`

, in terms of the order `...

**Pierre Hansen**

In the present paper, we compare the geometric-arithmetic index `\(GA\)`

and the chromatic number `\(\chi\)`

of a connected graph with given order. We prove,...

**Pierre Hansen**, Nenad Mladenović, Raca Todosijević, and Saïd Hanafi

Variable neighborhood search (VNS) is a framework for building heuristics, based upon systematic changes of neighborhoods both in a descent phase, to find a...

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In this paper, the first steps toward the use of the Variable Neighborhood Search metaheuristic are explained. The method is presented step by step using an...

BibTeX referenceSequential variable neighborhood descent variants: An empirical study on Travelling salesman problem

**Pierre Hansen**, and Nenad Mladenović

Usually several neighborhood structures may be explored within a single local search algorithm. The simplest way is to define a single neighborhood as a unio...

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In the present paper, we are interested in bounding differences between graph invariants as well as in characterizing the corresponding extremal graphs. This...

BibTeX reference**Pierre Hansen**

The distance signless Laplacian of a connected graph `\(G\)`

is defined by `\(\mathcal{D}^\mathcal{Q} = Diag(Tr) + \mathcal{D}\)`

, where `\(\mathcal{D}\)`

is...

**Pierre Hansen**

Proximity `\(\pi\)`

and remoteness `\(\rho\)`

are respectively the minimum and the maximum, over the vertices of a connected graph, of the average distance f...

**Pierre Hansen**

Finding communities in complex networks is a topic of much current research and has applications in many domains. On the one hand, criteria for doing so hav...

BibTeX reference**Pierre Hansen**, and Nenad Mladenović

The analysis of networks and in particular the identification of communities, or clusters, is a topic of active research with application arising in many dom...

BibTeX referenceDistance Spectra of Graphs: A Survey

**Pierre Hansen**

In 1971, Graham and Pollack established a relationship between the number of negative eigenvalues of the distance matrix and the addressing problem in data c...

BibTeX referenceLa recherche à voisinages variables

**Pierre Hansen**

La recherche à voisinages variables (RVV), ou <i>Variable Neighborhood Search (VNS)</i> en anglais est une métaheuristique dont l'invention est due à Nenad ...

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Variable neighborhood search (VNS) is a meta-heuristic for solving optimization problems, whose basic idea is a systematic change of neighborhood structure...

BibTeX reference**Pierre Hansen**

The distance Laplacian of a connected graph *G* is defined by *L = Diag(Tr) - D*, where *D* is the distance matrix of *G* , and *Diag(Tr)* is the diagonal m...

In the present paper, we are interested in studying mathematical properties of the Balaban index of connected graphs. We present a discussion on and refuta...

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Introduced during the late nineties of the last century, Variable Neighborhood Search (VNS) was first designed for solving specific problems in combinatorial...

BibTeX reference**Pierre Hansen**

We introduce a Laplacian and a signless Laplacian for the distance matrix of a connected graph, called the <i>distance Laplacian</i> and <i>distance signless...

BibTeX reference**Pierre Hansen**and Christophe Meyer

We derive conditions on the functions `\(\varphi\)`

, `\(\rho\)`

, `\(v\)`

and `\(w\)`

such that the 0-1 fractional programming problem`(\max\limits_{x\in {0...

Franchise Location Models and Cannibalization Effects: A Variable Neighborhood Search Approach

Application of the dispersion models in order to address the cannibalization phenomenon within franchised chains is a new approach. In this work we have deve...

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Community detection in networks has been studied extensively in the last decade. Many criteria, expressing the quality of the partitions obtained, as well ...

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Sequential clustering aims at determining homogeneous and/or well-separated clusters within a given set of entities, one at a time, until no more such clus...

BibTeX reference**Pierre Hansen**, Lucas Létocart, Leo Liberti, and Frédéric Messine

Reduced RLT constraints are a special class of Reformulation-Linearization Technique (RLT) constraints. They apply to nonconvex (both continuous and mixed-...

BibTeX reference**Pierre Hansen**

Given a set of entities, cluster analysis aims at finding subsets, also called clusters or communities or modules, entities of which are homogeneous and well...

BibTeX reference**Pierre Hansen**

Finding clusters, or communities, in a graph, or network is a very important problem which arises in many domains. Several models were proposed for its solu...

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The MaxSumSum (maximum diversity) problem consists of the selection of <i>p</i> facilities among <i>n</i> candidate locations in a way that the total sum ...

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Dispersion problems consist of the selection of a fixed number of vertices from a given set so that some function of the distances among the vertices is maxi...

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Since the late forties of the last century, methods of operations research have been extensively used to solve problems in graph theory, and graph theory has...

BibTeX referenceThe Small Octagons of Maximal Width

The paper answers an open problem introduced by Bezdek and Fodor in 2000. The width of any unit-diameter octagon is shown to be less than or equal to `(\fra...

BibTeX reference**Pierre Hansen**

Using a heuristic optimization module based upon Variable Neighborhood Search (VNS), the system AutoGraphiX's main feature is to find extremal or near extre...

BibTeX reference**Pierre Hansen**

We introduce a signless Laplacian for the distance matrix of a connected graph, called the <i>distance signless Laplacian</i>. We study the <i>distance signl...

BibTeX reference**Pierre Hansen**

We introduce a Laplacian for the distance matrix of a connected graph, called the <i>distance Laplacian</i> and we study its spectrum. We show the equivalenc...

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Finding communities, or clusters, or modules, in networks can be done by optimizing an objective function defined globally and/or by specifying conditions wh...

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Clusterwise regression is a clustering technique where multiple lines or hyperplanes are fit to mutually exclusive subsets of a dataset such that the sum of ...

BibTeX reference**Pierre Hansen**

In 1956, Nordhaus and Gaddum gave lower and upper bounds on the sum and the product of the chromatic number of a graph and its complement, in terms of the or...

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Modularity maximization is extensively used to detect communities in complex networks. It has been shown however that this method suffers from a resolution l...

BibTeX reference**Pierre Hansen**, and Leo Liberti

We study the problem of packing equal circles in a square from the mathematical programming point of view. We discuss different formulations, we analyse fo...

BibTeX reference**Pierre Hansen**

In a recent paper, Zhan, Zhang, Guan, and Zhou [Phys. Rev. E <b>83</b>, 066120 (2011)] presented a modified adaptive genetic algorithm (MAGA) tailored to the...

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Finding communities, or clusters, in networks, or graphs, has been the subject of intense studies in the last ten years. The most used criterion for that pu...

BibTeX referenceThe Normalized Revised Szeged Index

**Pierre Hansen**

In chemical graph theory, many graph parameters, or topological indices, were proposed as estimators of molecular structural properties. Often several varian...

BibTeX referenceDegeneracy of Harmonic Means Clustering

**Pierre Hansen**, and Nenad Mladenović

It is well known that some local search algorithms for <i>K</i>-clustering problems could stop at a solution with fewer clusters than the desired <i>K</i>....

BibTeX referenceMaximizing Edge-Ratio Is NP-Complete

**Pierre Hansen**, and Nenad Mladenović

Given a graph <i>G</i> and a bipartition of its vertices, the edge-ratio is the minimum for both classes so defined of their number of internal edges divid...

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The objective in the continuous facility location problem with limited distances is to minimize the sum of distance functions from the facility to the cust...

BibTeX reference**Pierre Hansen**, Christophe Meyer, and L-N Tellier

The GRIEG model is a hybrid model of demo-economic projections that combines two approaches: the econometric approach - based on the micro-economy - of the N...

BibTeX reference**Pierre Hansen**, and Leo Liberti

Community detection in networks based on modularity maximization is currently done with hierarchical divisive or agglomerative as well as with partitioning h...

BibTeX reference**Pierre Hansen**, and Leo Liberti

Heuristics are widely applied to modularity maximization models for the identification of communities in complex networks. We present an approach to be appli...

BibTeX referenceExtensions to the Repetitive Branch and Bound Algorithm for Globally Optimal Clusterwise Regression

A branch and bound strategy is proposed for solving the clusterwise regression problem, extending Brusco's repetitive branch and bound algorithm (RBBA). The ...

BibTeX reference**Pierre Hansen**and Christophe Meyer

We present a new column generation algorithm for the determination of a classifier in the two classes LAD (Logical Analysis of Data) model. Unlike existing a...

BibTeX reference**Pierre Hansen**, and Claire Lucas

We study extremal graphs for the extremal values of the second largest <i>Q</i>-eigenvalue of a connected graph. We first characterize all simple connected g...

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Normalized cut is one of the most popular graph clustering criteria. The main approaches proposed for its resolution are spectral clustering methods (e.g. [1...

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Finding modules, or clusters, in networks currently attracts much attention in several domains. The most studied criterion for doing so, due to Newman and Gi...

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An automatic method for constructing linear relaxations of constrained global optimization problems is proposed. Such a construction is based on affine and i...

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Exact global optimization of the clusterwise regression problem is challenging and there are currently no published feasible methods for performing this clus...

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In this paper we establish the definition of the <i>set of ε-proper equilibria</i> of a bimatrix game. We define a 0-1 mixed quadratic program to ge...

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The main goal of this paper is to bring a contribution in order to facilitate automatic refinement of Bimatrix Game Nash extreme equilibria. We show how maxi...

BibTeX referenceA Note on Diameters of Point Sets

Relationships between the diameter of a set of <i>n</i> points in the plane at mutual distance at least one, the diameter of an equilateral <i>n</i>-gon and ...

BibTeX reference**Pierre Hansen**, and Leo Liberti

The modularity maximization model proposed by Newman and Girvan for the identification of communities in networks works for general graphs possibly with loop...

BibTeX reference**Pierre Hansen**, and Leo Liberti

A new hierarchical divisive algorithm is proposed for identifying communities in complex networks. To that effect, the definition of <i>community in the weak...

BibTeX reference**Pierre Hansen**and Claire Lucas

Using the AutoGraphiX system, we obtain conjectures of the form <img src="/cgi-bin/mimetex.cgi?l(n) \leq q_1 \oplus i(G)\leq u(n)"> where <img src="/cgi-bin...

BibTeX reference**Pierre Hansen**, and Dragan Stevanovic

Let <i>G</i> be a connected graph of order <i>n</i>. The algebraic connectivity of <i>G</i> is the second smallest eigenvalue of the Laplacian matrix of <i>G...

BibTeX reference**Pierre Hansen**

The <i>proximity</i> <img src="/cgi-bin/mimetex.cgi?\pi = \pi (G)"> of a connected graph <i>G</i> is the minimum, over all vertices, of the average distance ...

BibTeX reference**Pierre Hansen**

We address the problem of locating in the plane objects such as segments, arcs of circumferences, arbitrary convex sets, their complements or their boundarie...

BibTeX reference**Pierre Hansen**, Nenad Mladenović, and Eric W.T. Ngai

Harmonic means clustering is a variant of Minimum sum of squares clustering (which is sometimes called <i>K</i>-means clustering), designed to alleviate the ...

BibTeX referenceVariable Neighborhood Search

Variable neighborhood search (VNS) is a metaheuristic for solving combinatorial and global optimization problems whose basic idea is a systematic change o...

BibTeX reference**Pierre Hansen**and Claire Lucas

Let <i>Q = D + A</i> denote the signless Laplacian matrix of a graph <i>G</i> of order <i>n</i>, where <i>D</i> is the diagonal matrix of the degrees and <i...

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Given a graph <i>G=(V,E)</i>, the first Zagreb index <i>M</i><sub>1</sub> is the sum of its vertices squared degrees and the second Zagreb index <i>M</i><sub...

BibTeX referenceOn a Conjecture About the Szeged Index

**Pierre Hansen**

Khalifeh, Yousefi-Azari, Ashrafi and Wagner [European J. Combin. 30 (2009) 1149-1163] conjectured that for a connected graph <i>G</i> on <i>n</i> vertices...

BibTeX reference**Pierre Hansen**

The proximity <img src="/cgi-bin/mimetex.cgi?\pi"> of a graph <i>G</i> is the minimum average distance from a vertex of <i>G</i> to all others. Similarly, th...

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Given a set of entities associated with points in Euclidean space, minimum sum-of-squares clustering (MSSC) consist in partitioning this set into clusters su...

BibTeX reference**Pierre Hansen**

During the last three decades, the computer has been widely used in spectral graph theory. Many results about graph eigenvalues were first conjectured, and i...

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It is observed that mutations in the formulations of test problems over time are not infrequent. Ensuing problems are illustrated with examples from geometri...

BibTeX reference**Pierre Hansen**, and Abdelwaheb Rebaï

Finding maximum likelihood parameter values for Finite Mixture Model (FMM) is often done with the Expectation Maximization (EM) algorithm. However the choice...

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A recent paper of Tuy and Hoai-Phuong published in <i>JOGO</i> (2007) 37:557--569 observes that there are errors in the reformulation of a signomial geometr...

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Necessary and sufficient conditions are provided for existence of a simple graph <i>G</i>, and for a simple and connected graph <i>G'</i> with given numbers ...

BibTeX reference**Pierre Hansen**

Upper bounds on the average distance `\(\overline{l}\)`

between pairs of vertices of a connected graph with given order `\(n\)`

and minimum degree `(\delta...

**Pierre Hansen**

The transmission of a vertex in a connected graph is the sum of all distance from that vertex to the others. It is said to be normalized if divided by <i>n-1...

BibTeX reference**Pierre Hansen**, Nenad Mladenović, and José A. Moreno Pérez

Variable neighborhood search (VNS) is a metaheuristic, or framework for building heuristics, based upon systematic change of neighborhoods both in a decent ...

BibTeX reference**Pierre Hansen**

Minimum sum-of-squares clustering (MSSC) consists in partitioning a given set of <i>n</i> points into <i>k</i> clusters in order to minimize the sum of squar...

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A polygon is said to be <i>simple</i> if the only points of the plane belonging to two of its edges are its vertices. We answer the question of finding, ...

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With the help of the Graffiti system, Fajtlowicz conjectured around 1992 that the average distance between two vertices of a connected graph <i>G</i> is at ...

BibTeX reference**Pierre Hansen**

Methods, models, heuristic and exact algorithms for clustering are reviewed from a mathematical programming view point.

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A recent proof of NP-hardness of Euclidean sum-of-squares clustering, due to Drineas et al., <i>Machine Learning</i> 56, 9--33, 2004, is not valid. An altern...

BibTeX reference**Pierre Hansen**

Let <img src="/cgi-bin/mimetex.cgi?G=(V,E)"> be a simple, undirected graph of order <img src="/cgi-bin/mimetex.cgi?n"> and size <img src="/cgi-bin/mimetex.cg...

BibTeX reference**Pierre Hansen**

In this note we suggest a simple but efficient modification of the well-known Nelder-Mead (NM) simplex search method for unconstrained optimization. Instead ...

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The hexagon and heptagon with unit diameter and maximum sum of Euclidean distances between vertices are determined by enumerating diameter configurations, an...

BibTeX reference**Pierre Hansen**

Minimum sum-of-squares clustering consists in partitioning a given set of <i>n</i> points into <i>c</i> clusters in order to minimize the sum of squared dist...

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We consider one of the most important issues for multinationals, the determination of transfer prices. To do so, we examine the example of a multinational co...

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In this paper we present a simple technique that uses background information to improve mining the frequent patterns of structured data. This technique uses ...

BibTeX referenceSyGMA: Reducing Symmetry in Graph Mining

While recent algorithms for mining the frequent subgraphs of a database are efficient in the general case, these algorithms tend to do poorly on databases th...

BibTeX reference**Pierre Hansen**

With the help of the AutoGraphiX system, we study relations of the form

<img src="/cgi-bin/mimetex.cgi?\underline{b}*m \le i*1(G) \oplus i_2(G) \le \overl...

**Pierre Hansen**

Clusterwise regression is a technique for clustering data. Instead of using the classical homogeneity or separation criterion, clusterwise regression is ba...

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In previous work, bimatrix games were expressed as parametric linear mixed 0-1 programs and the <img src="/cgi-bin/mimetex.cgi?{\rm E}_\chi">MIP algorithm w...

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Consider a convex polygon <i>V<sub>n</sub></i> with <i>n</i> sides, perimeter <i>P<sub>n</sub></i>, diameter <i>D<sub>n</sub></i>, area <i>A<sub>n</sub></i>...

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The AutoGraphiX system (AGX 1 and AGX 2) for interactive, and for several functions automated, graph theory, discovers conjectures of algebraic or structura...

BibTeX reference**Pierre Hansen**

<p>On étudie à l'aide du système AutoGraphiX 2 (AGX 2) des relations de la forme </p>

<p><center> <img src="/cgi-bin/mimetex.cgi?\underline{b}_{n} \, \l...

BibTeX referenceVariable Neighborhood Search Methods

**Pierre Hansen**and Nenad Mladenović

Main methods, algorithms and applications of the Variable Neighborhood Search metaheuristic are surveyed, in view of a chapter of the <i>Encyclopedia of Opti...

BibTeX reference**Pierre Hansen**

To the best of our knowledge, the complexity of minimum sum-of-squares clustering is unknown. Yet, it has often been stated that this problem is NP-hard. We...

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Given a class of graphs <img src="/cgi-bin/mimetex.cgi?\mathcal{F}">, a forbidden subgraph characterization (FSC) is a set of graphs <img src="/cgi-bin/mimet...

BibTeX referenceVariable Neighborhood Search for Extremal Graphs. 25. Products of Connectivity and Distance Measures

**Pierre Hansen**

Upper bounds for products of four measures of distances in graphs: diameter, radius, average eccentricity and remoteness with three measures of connectivity:...

BibTeX reference**Pierre Hansen**

In the set of all connected graphs with a given domination number, we characterize the graphs which achieve the maximum value of the spectral radius of the ...

BibTeX reference**Pierre Hansen**

We show that among connected graphs with maximum clique size <img src="/cgi-bin/mimetex.cgi?\omega">, the minimum value of the spectral radius of adjacency m...

BibTeX reference**Pierre Hansen**and Christophe Meyer

We present and compare three new compact linearizations for the quadratic 0-1 minimization problem, two of which achieving the same lower bound than the "sta...

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Data clustering methods have been developed extensively in the data mining literature for detecting useful patterns in large datasets in the form of dense...

BibTeX reference**Pierre Hansen**

A set of vertices <i>S</i> in a graph <i>G</i> is a clique if any two of its vertices are adjacent. The clique number <img src="/cgi-bin/mimetex.cgi?\omega"...

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In this survey, we examine an important class of facility location problems known as the multisource Weber problem (also referred to as the continuous locat...

BibTeX reference**Pierre Hansen**and Nenad Mladenović

A recent comparison of evolutionary, neural network, and scatter search heuristics for solving the p-median problem is completed by (i) gathering or obtaini...

BibTeX reference**Pierre Hansen**and Sylvain Perron

The probabilistic satisfiability problem is to verify the consistency of a set of probability values or intervals for logical propositions. The (tight) prob...

BibTeX referenceExact

*L*

_{2}-Norm Plane Separation

We consider the problem of separating two sets of points in an n-dimensional real space with a (hyper)plane that minimizes the sum of <i>L</i><sub>2</sub>-n...

BibTeX reference**Pierre Hansen**

A set of vertices <i>S</i> in a graph <i>G</i> is independent if no neighbor of a vertex of <i>S</i> belongs to <i>S</i>. A set of vertices <i>U</i> in a gr...

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This paper presents two new results on the enumeration of all extreme equilibria of the sequence form of a two person extensive game. The sequential form of...

BibTeX reference**Pierre Hansen**, and Dragan Stevanovic

The AutoGraphiX 2 system is used to compare the index of a graph <i>G</i> with a number of other graph theoretical invariants, i.e., chromatic number, maxim...

BibTeX referenceVariable Neighborhood Search for Extremal Graphs. 20. Automated Comparison of Graph Invariants

A graph invariant is a function of a graph <i>G</i> which does not depend on labeling of <i>G</i>’s vertices or edges. An algebraic expression of one or sev...

BibTeX reference**Pierre Hansen**and Sylvain Perron

Assouad has shown that a real-valued distance <i>d</i> = (<i>d</i><sub><i>ij</i></sub>) <sub>1 ≤ <i>i</i> < <i>j</i> ≤ <i>n</i></sub> is isome...

BibTeX referenceIsoperimetric Polygons of Maximal Width

The value <img src="/cgi-bin/mimetex.cgi?\frac{1}{2n} \cot\left( \frac{\pi}{2n}\right)"> is shown to be an upper bound on the width of any <i>n</i>-sided p...

BibTeX reference**Pierre Hansen**

A set of vertices <i>S</i> in a graph <i>G</i> is independent if no neighbor of a vertex of <i>S</i> belongs to <i>S</i>. The independence number &alpha is ...

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From the pentagon onwards, the area of the regular convex polygon with <i>n</i> sides and unit diameter is greater for each odd number <i>n</i> than for the...

BibTeX reference**Pierre Hansen**, Peter Rowlinson, S Simic, and Dragan Stevanovic

We consider four conjectures related to the largest eigenvalue of (the adjacency matrix of) a graph (i.e., to the index of the graph). Three of them have be...

BibTeX referenceVariable Neighborhood Search for Extremal Graphs. 23. On the Randic Index and the Chromatic Number

**Pierre Hansen**and Damir Vukicevic

Let
<img src="/cgi-bin/mimetex.cgi?G=(V,E)"> be a simple graph with vertex degrees
<img src="/cgi-bin/mimetex.cgi?d*1, d*2, \ldots, d_n...

Comparing the Zagreb Indices

**Pierre Hansen**and Damir Vukicevic

Let <img src="/cgi-bin/mimetex.cgi?G=(V,E)"> be a simple graph with <img src="/cgi-bin/mimetex.cgi?n=|V|"> vertices and <img src="/cgi-bin/...

BibTeX reference**Pierre Hansen**and Nikolaj van Omme

Given a simple connected graph <i>G = (V,E)</i> the geodetic closure <i>I [S]</i> <img src="/cgi-bin/mimetex.cgi?\subset"> <i>V</i> of a subset <i>S</i...

BibTeX referenceThe Jackknife Revisited: Feature Selection in Linear Programming Approaches to Credit Scoring

**Pierre Hansen**

We discuss feature selection approaches within the context of linear programming models for discrimination, with special emphasis on a new interpretation an...

BibTeX reference**Pierre Hansen**, and Maolin Zheng

The AutoGraphiX research program led to a new type of application of metaheuristics in graph theory, <i>i.e.</i>, finding conjectures on graph invariants by...

BibTeX referenceVariable Neighborhood Search for Extremal Graphs. 18. Conjectures and Results about the Randic Index

**Pierre Hansen**, and Maolin Zheng

Using the AutoGraphiX 2 system (AGX2), we study relations between graph invariants
of the form
<img src="/cgi-bin/mimetex.cgi?lb*n \leq R \oplus i \leq ub*...

**Pierre Hansen**, Jasmina Lazic, and Nenad Mladenović

Colour image quantization is a data compression technique that reduces the total set of colours in an image to a representative subset. This problem is expr...

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We consider the problem of separating two sets of points in an Euclidean space with a hyperplane that minimizes the sum of <i>L<sub>p</sub></i>-norm distanc...

BibTeX referenceExtremal Problems for Convex Polygons

Consider a convex polygon <i>V<sub>n</sub></i> with <i>n</i> sides, perimeter <i>P<sub>n</sub></i>, diameter <i>D<sub>n</sub></i>, area <i>A<sub>n</sub></i>,...

BibTeX referenceQuatre Petits Octogones

Quel octogone de diamère unité (ou petit octogone) possède la plus grande surface ou le plus grand périmètre? Serait-ce l'octogone régulier? Eh! non, il n'en...

BibTeX referenceOn a Conjecture About the Randic Index

**Pierre Hansen**

A conjecture of Delorme, Favaron and Rautenbach [DM 257 (2002) 29-38] about the Randic index of a graph, in relation to its order and minimum degree, is ref...

BibTeX reference**Pierre Hansen**

<p>Le système <i>AutoGraphiX (AGX1 et AGX2)</i> permet, parmi d’autres fonctions, la génération automatique de conjectures en théorie des graphes. Nous étud...

BibTeX reference**Pierre Hansen**

<p>Using the <i>AutoGraphiX 2</i> system, a systematic study is made on generation and proof of relations of the form</p> <center> $\underline{b}_n \leq ...

BibTeX referenceSet covering and packing formulations of graph coloring: algorithms and first polyhedral results

**Pierre Hansen**, Martine Labbé, and David Schindl

We consider two (0,1)-linear programming formulations of the graph (vertex-) coloring problem, in which variables are associated to stable sets of the input...

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<i>L</i><sub>1</sub> norm discrimination consists in finding the hyperplane that minimizes the sum of <i>L</i><sub>1</sub> norm distances between the hyperp...

BibTeX referenceThe Small Octagon with Longest Perimeter

The convex octagon with unit diameter and maximum perimeter is determined. This answers an open question dating from 1922. The proof uses geometric reasonin...

BibTeX referenceAutoGraphiX: A Survey

A survey is made of the AutoGraphiX (AGX) research program for computer as- sisted and, for some functions, automated graph theory.

BibTeX reference**Pierre Hansen**and Dragan Stevanovic

Usual graph classes, such as complete graphs, paths, cycles and stars, frequently appear as extremal graphs in graph theory problems. Here we want to turn t...

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The <i>p</i>-median problem is one of the basic models in discrete location theory. As with most location problems, it is classified as NP-hard, and so, heu...

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The variable neighborhood search metaheuristic is applied to the primal simple plant location problem and to a reduced dual obtained by exploiting the compl...

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This paper examines the plant location problem under the objective of maximizing return-on-investment. However, in place of the standard assumption that all...

BibTeX reference**Pierre Hansen**, and Dragan Stevanovic

Several upper bounds on the largest Laplacian eigenvalue of a graph <i>G</i>, in terms of degree and average degree of neighbors of its vertices, have been ...

BibTeX reference**Pierre Hansen**, L Hiesse, J Lacheré, and A Monhait

The AutoGraphiX (AGX) system for computer assisted or, for some of its functions, fully automated graph theory was developed at GERAD, Montreal since 1997. ...

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The continuous location-allocation problem requires finding sites for <i>m</i> new facilities in the plane in order to serve <i>n</i> users such that the to...

BibTeX referenceParallel Variable Neighborhood Search

**Pierre Hansen**, and Nenad Mladenović

Variable Neighborhood Search (VNS) is a recent and effective metaheuristic for solving combinatorial and global optimization problems. It is capable of esca...

BibTeX reference**Pierre Hansen**

Les problèmes de chargement statique de groupes turbo-alternateurs à vapeur peuvent être exprimés sous forme de minimisation d’une somme de fonction de coût ...

BibTeX reference**Pierre Hansen**

Dynamic programming is applied to the economic dispatch problem with spin- ning reserve constraint. A first algorithm, based on a direct application of Bell...

BibTeX referenceAnalysis of Global

*k*-Means, an Incremental Heuristic for Minimum Sum-of-Squares Clustering

**Pierre Hansen**, Eric W.T. Ngai, Bernard K.-S. Cheung, and Nenad Mladenović

The global <i>k</i>-means heuristic is a recently proposed (Likas et al. 2003) incremental approach for minimum sum-of-squares clustering of a set <i>X</i> ...

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We consider the problem of separating two sets of points in an <i>n</i>-dimensional real space with a (hyper)plane that minimizes the sum of <i>L<sub>p</sub...

BibTeX reference**Pierre Hansen**, Nenad Mladenović, and Dragan Urosević

In this paper we develop a Variable neighborhood search (VNS) heuristic for solving Mixed-Integer Programs (MIP’s). It uses CPLEX, the general-purpose MIP s...

BibTeX reference**Pierre Hansen**, Hadrien Mélot, and Ivan Gutman

An upper bound is given on the variance of degrees of graphs with <i>n</i> vertices, <i>m</i> edges and maximum degree Δ. Particular cases of chemical...

BibTeX reference**Pierre Hansen**, and Carla Silva Oliveira

The algebraic connectivity <i>a(G)</i> of a graph <i>G = (V,E)</i> is the second smallest eigenvalue of its Laplacian matrix. Using the AutoGraphiX (AGX) sys...

BibTeX reference**Pierre Hansen**, and Hadrien Mélot

Chemical graphs, as other ones, are <i>regular</i> if all their vertices have the same degree. Otherwise, they are <i>irregular</i> and it is of interest to...

BibTeX reference**Pierre Hansen**, and Vera Kovacevic-Vujcic

The uniting feature of combinatorial optimization and extremal graph theory is that in both areas one should ﬁnd extrema of a function defined in most...

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Conjectures in graph theory have multiple forms and involve graph invariants, graph classes, subgraphs, minors and other concepts in premisses and/or conclu...

BibTeX reference**Pierre Hansen**

Computer-assisted and automated conjecture-making in graph theory is reviewed, focusing on the three operational systems GRAPH, Graffiti and AutoGraphiX (AGX...

BibTeX reference**Pierre Hansen**, Jean-Louis Lagouanelle, and Frédéric Messine

Two ways for bounding <i>n</i>-variables functions over a box, based on interval evalu- ations of first order derivatives, are compared. The optimal Baumann...

BibTeX reference**Pierre Hansen**, Yuri Kochetov, and Nenad Mladenović

We consider the bilevel uncapacitated facility location problem with user preferences. It is known that this model may be reformulated as a one-level locati...

BibTeX reference**Pierre Hansen**, and Nenad Mladenović

The multiprocessor scheduling problem with communication delays that we consider in this paper consists of finding a static schedule of an arbitrary task gra...

BibTeX reference**Pierre Hansen**

In this exploratory study, a segmentation analysis of a shopping mall’s customers is conducted according to the activities they performed during their visit,...

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This paper examines the convergence properties of two well-known heuristics: variable neighborhood search (VNS) and random multistart local search (MLS). Bot...

BibTeX reference**Pierre Hansen**and Nenad Mladenović

When applying the 2-opt heuristic to the travelling salesman problem, selecting the best improvement at each iteration gives worse results on average than se...

BibTeX reference**Pierre Hansen**, C Oguz, and Nenad Mladenović

The berth allocation problem is to allocate space along the quayside to incoming ship at a container terminal in order to minimize some objective function....

BibTeX reference**Pierre Hansen**and C Oguz

The berth allocation problem is to allocate berths (i.e., sections of the quayside) to ships arriving in a container port in order to minimize the sum of the...

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Bimatrix and polymatrix games are expressed as parametric linear 0-1 programs. This leads to an algorithm for complete enumeration of their extreme equilibri...

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Goal Programming with fractional objectives can be reduced to mathematical programming with a linear objective under linear and quadratic constraints, thus...

BibTeX referenceThe Minimum Diameter Octagon with Unit-Length Sides: Vincze's Wife's Octagon is Suboptimal

This paper answers a query of S. Vincze (Acta Univ. Szeged, Sect. Sci. Math. 12 A (1950) 136-142): find the convex octagon with unit-length sides and minim...

BibTeX reference**Pierre Hansen**and Nenad Mladenović

Variable Neighborhood Search (VNS) is a recent metaheuristic, or framework for building heuristics, which exploits systematically the idea of neighborhood ch...

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We define two classes of lower bounds using either one or two simplices for the minimization of a concave function over a polytope. For each of them, a pro...

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Consider <i>N</i> entities to be classified (e.g., geographical areas), a matrix of dissimilarity between pairs of entities, a graph <i>H</i> with vertices a...

BibTeX reference**Pierre Hansen**

It is well known that the set of correlated equilibrium distributions of a noncooperative game is a convex polytope that includes all the Nash equilibrium ...

BibTeX reference**Pierre Hansen**, and Frédéric Messine

We explore how a simple linear change of variable affects the inclusion functions obtained with Interval Analysis methods. Univariate and multivariate pol...

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We present an exact algorithm and three applications of nonconvex quadratically constrained quadratic programming. First, we consider the pooling problem fro...

BibTeX reference**Pierre Hansen**, and Nenad Mladenović

We develop and analyze parallelization strategies for the Variable Neighborhood Search (VNS) meta-heuristic applied to the <i>p</i>-median problem. The stra...

BibTeX reference**Pierre Hansen**and Hadrien Mélot

Albertson (1997) defines the <i>imbalance</i> of an edge (<i>i,j</i>) in <i>E</i> of a graph <i>G</i> = (<i>V,E</i>) as | <i>d<sub>i</sub> - d<sub>j</sub></...

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This paper studies the duality gap in the simple plant location problem, and presents general formulas for the gap when certain complementary slackness cond...

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After a short historic review, we briefly describe a new algorithm for constructive enumeration of polyhex and fusene hydrocarbons. In this process our alg...

BibTeX reference**Pierre Hansen**, and Martine Labbé

Generalized logical analysis aims at modelling complex biological systems, especially the so-called regulatory systems like genetic networks. The main feat...

BibTeX referenceVariable Neighborhood Search for Extremal Graphs. 6. Analyzing Bounds for the Connectivity Index

**Pierre Hansen**and Hadrien Mélot

Recently, Araujo and De la Pena (1998) gave bounds for the connectivity index of chemical trees as a function of this index for general trees and the ramifi...

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The pooling problem, which is fundamental to the petroleum industry, describes a situation where products possessing different attribute qualities are mixed...

BibTeX reference**Pierre Hansen**, and Dragan Stevanovic

We prove that amongst all fullerenes the dodecahedron has maximum smallest eigenvalue (equal to -<img src="G0234-1.gif" align=middle>), followed by the thre...

BibTeX referenceVariable Neighborhood Search for Extremal Graphs: 5. Three Ways to Automate Finding Conjectures

**Pierre Hansen**

The AutoGraphiX (AGX) system determines classes of extremal or near extremal graphs with a Variable Neighborhood Search heuristic. From these, conjectures m...

BibTeX referenceComputers in Graph Theory

**Pierre Hansen**

A survey is made of computer systems which help to obtain and sometimes provide automatically proofs, conjectures and refutations in graph theory.

BibTeX referenceOn Uniform

*k*-Partition Problems

**Pierre Hansen**, S Pallottino, and G Storchi

We study various uniform <i>k</i>-partition problems which consist in partitioning <i>m</i> sets, each of cardinality <i>k</i>, into <i>k</i> sets of cardin...

BibTeX referenceGraphs with Maximum Connectivity Index

Let <i>G</i> be a graph and <i>d<sub>v</sub></i> the degree (= number of first neighbors) of its vertex <i>v</i>. The connectivity index of <i>G</i> is <img...

BibTeX reference**Pierre Hansen**and Hadrien Mélot

We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory.

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In this paper, a fast and complete method to constructively enumerate fusenes and benzenoids is given. It is fast enough to construct several million non is...

BibTeX referencePanchromatic Chains and Paths

**Pierre Hansen**

A generalization of the Roy-Gallai theorem on the chromatic number of a graph is derived which is also an extension of several other results of Berge and of...

BibTeX referenceIntegral Complete Split Graphs

**Pierre Hansen**, Hadrien Mélot, and Dragan Stevanovic

We give characterizations of integral graphs in the family of complete split graphs and a few related families of graphs.

BibTeX reference**Pierre Hansen**and Nenad Mladenović

Variable neighborhood search (VNS) is a recent metaheuristic for solving combinatorial and global optimization problems whose basic idea is systematic chang...

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Graffiti's conjecture 105 states that for any tree the range of transmissions of distance is greater than or equal to the range of degrees. After using the ...

BibTeX referenceThe Largest Small Octagon

Thrackleation of graphs and global optimization for quadratically constrained quadratic programming are used to find the octagon with unit diameter and larg...

BibTeX reference**Pierre Hansen**and Nenad Mladenović

Proposed just a few years ago, Variable Neighborhood Search (VNS) is a new metaheuristic for combinatorial and global optimization, based upon systematic ch...

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Reduction of some classes of global optimization programs to bilinear programs may be done in various ways, and the choice of method clearly influences the ...

BibTeX reference**Pierre Hansen**

The <i>p</i>-Center problem consists in locating <i>p</i> facilities and assigning clients to them in order to minimize the maximum distance between a clien...

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On the basis of a variable-neighbourhood search with the AutoGraphiX software, it is conjectured that for even numbers of atoms the fully conjugated acycli...

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A fuzzy clustering problem consists in assigning a set of patterns to a given number of clusters with respect to some criteria such that each of them may be...

BibTeX referenceStable Sets and Chromatic Number

**Pierre Hansen**

A generalization of the Roy-Gallai theorem is presented: it is based on the existence in any oriented graph of a stable set S such that for any node <i>w</i...

BibTeX reference**Pierre Hansen**, Nenad Mladenović, and Dragan Urosević

Maximum Clique is one of the most studied NP-hard optimization problem on graphs because of its simplicity and its numerous applications. A basic Variable N...

BibTeX reference**Pierre Hansen**, and Federico Malucelli

An algorithm with a complexity linear in the number of vertices is proposed for the computation of the Hyper-Wiener index of chemical trees. This complexity...

BibTeX referenceRecherche à Voisinage Variable

**Pierre Hansen**and Nenad Mladenović

La Recherche à Voisinage Variable (RVV) est une métaheuristique récente basée sur l'idée d'un chargement systématique de voisinage, à la fois dans une phase...

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The distance matrix of a chemical graph can be computed in quadratic time, and from it can be obtained the distance level patterns (DLP), Wiener, Szeged and...

BibTeX referenceAn Oil Pipeline Design Problem

We consider a given set of offshore platforms and onshore wells, producing known (or estimated) amounts of oil, to be connected to a port. Connections may t...

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Given a set of logical sentences and probabilities that these sentences are true, the probabilistic logic problem consists in determining whethe...

BibTeX reference**Pierre Hansen**and Nenad Mladenović

Variable Neighborhood Search (VNS) is a recent metaheuristic which exploits systematically the idea of change of neighborhood within the search. After recal...

BibTeX referenceConnectivity, Transitivity and Chromaticity: The Pioneering Work of Bernard Roy in Graph Theory

**Pierre Hansen**and Dominique De Werra

Given a set of logical sentences together with nonnegative weights assigned to each of them, the Maximum Weight Satisfiability problem (MAX-WEIGHT SAT) cons...

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We modify the algorithm of Pardalos and Rodgers [40] for the minimization of a pseudo-boolean quadratic function by introducing an easy to compute lower bou...

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The set of equilibrium points of a bimatrix game is the union of polytopes that are not necessarily disjoint. Knowledge of the vertices of these polytopes ...

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Clique partitionning in Euclidean space <b>R</b><sup>n</sup> consists in finding a partition of a given set of <i>N</i> points into <i>M</i> clusters in ord...

BibTeX reference**Pierre Hansen**and Nenad Mladenović

The standard way to solve the static economic dispatch problem with transmission losses is the penalty factor method. The problem is solved iteratively by a...

BibTeX reference**Pierre Hansen**and Nenad Mladenović

Systematic change of neighborhood within a possibly randomized local search algorithm yields a simple and effective metaheuristic for combinatorial and glo...

BibTeX reference**Pierre Hansen**, Stéphane Krau, Dominique Peeters, and Jacques-Francois Thisse

Weber's problem is to locate a facility in the Euclidean plane in order to minimize total transportation costs from that facility to a given set of users wi...

BibTeX reference**Pierre Hansen**

A model for the optimal location of new facilities in a competitive market is introduced under the hypothesis that customers' behavior can be modeled by ran...

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In the set of bicolored trees with given numbers of black and of white vertices we describe those for which the largest eigenvalue is extremal (maximal or m...

BibTeX referenceRegards sur la découverte

**Pierre Hansen**

Quatre regards sur la découverte, ceux du psychologue, de l'historien, du sociologue et du cognicien sont évoqués en quelques remarques illustrées d'exemples.

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Mirkin's additive clustering (or qualitative factor analysis) algorithm explains similarities between entities by sequentially finding a cluster of entities...

BibTeX referenceBoundary Uniqueness of Fusenes

**Pierre Hansen**, and Maolin Zheng

It is shown that a geometrically planar fusene is uniquely determined by its boundary edge code. Surprisingly, the same conclusion is not true in general b...

BibTeX reference**Pierre Hansen**and Nenad Mladenović

Variable Neighborhood Search is a recent metaheuristic based on the idea of systematic change of neighborhood during both a descent phase and an exploration...

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The definition of a fullerene as a cubic polyhedron made up entirely of pentagons and hexagons is compatible with a huge variety of isomeric forms for struc...

BibTeX reference**Pierre Hansen**and Nenad Mladenović

A new local search heuristic, called J-MEANS, is proposed for solving the minimum sum-of-squares clustering problem. The neighborhood of the current solut...

BibTeX referenceEnumeration of Fusenes to

*h*= 20

Fusenes are simply connected, geometrically planar or non-planar polyhexes. Enumeration of such systems with up to 20 hexagons, <i>i.e.,</i> a set 4000 tim...

BibTeX reference**Pierre Hansen**, Nenad Mladenović, and D Perez-Brito

The recent Variable Neighborhood Search (VNS) metaheuristic combines local search with systematic changes of neighborhood in the descent and escape from loc...

BibTeX reference**Pierre Hansen**

Let <i>G</i> be a simple graph and <i>C</i> and <i>D</i> two proper colourings of <i>G</i>. The problem of colour switching consists of finding a sequence ...

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Treatment of imprecise probabilities within the probabilistic satisfiability approach to uncertainty in knowledge-based systems is surveyed and discussed. ...

BibTeX reference**Pierre Hansen**

Given a set of <i>m</i> observations on <i>n</i> variables, an 0(<i>mn</i><sup>2</sup>) algorithm is proposed to find a basis of all affine relations betwee...

BibTeX referenceVariable Neighborhood Search for Extremal Graphs. 4. Chemical Trees with Extremal Connectivity Index

By means of the variable neighborhood search algorithm, a newly designed heuristic approach to combinatorial optimization, we established the structure of t...

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The probabilistic satisfiability problem consists, given <i>m</i> logical sentences, with their probabilities to find if they are coherent, and if it is the...

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We discuss the relationship between probabilistic logic and <img src="pi.gif" align=bottom>-CMS. Given a set of logical sentences and probabilities, the ou...

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Given a network modeled by a probabilistic graph <i>G =</i> (<i>V,E</i>) with bounds on the operation probabilities of edges and of pairs of edges, the seco...

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The recently developed Variable Neighborhood Search (VNS) meta-heuristic for combinatorial and global optimization is outlined together with its specializat...

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The chemical trees on <i>n</i> vertices (= molecular graphs representing alkanes with <i>n</i> carbon atoms) with minimal, second-minimal, third-minimal, ma...

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Let <i>G</i> be a connected graph and let <i>D</i> be its diameter. Denote by <i>d</i>(<i>G,k</i>) the number of pairs of vertices in <i>G</i> that are at d...

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We present a branch and cut algorithm that yields in finite time, a globally <img src="epsilon.gif" align=bottom>-optimal (with respect to feasibility and o...

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An exact algorithm is proposed for minimum sum-of-squares nonhierarchical clustering, i.e., for partitioning a given set of points from Euclidean <i>m</i>-s...

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Given a graph <i>G</i> with <i>n</i> vertices and stability number <img src="alpha.gif" align=bottom>(<i>G</i>), Turán's theorem gives a lower bound on the ...

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We study empirically the topology of the local minima of the 3-SAT problem. In particular, we analyze the size of the plateaus, their altitude, their attrac...

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We present branch and bound algorithms that enumerate in finite time all Nash equilibria for strategic and sequence form bimatrix games. For each forms, th...

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We consider Nash equilibria as correlated equilibria and apply polyhedral theory to study extreme Nash equilibrium properties. We obtain an alternate proof ...

BibTeX reference**Pierre Hansen**, KM Rogers, and Siemion Fajtlowicz

If it is assumed that the final product of bromination of C<sub>60</sub> will obey two rules, (i) that no two <i>sp</i><sup>3</sup> carbons may be adjacent,...

BibTeX referenceStabilized Column Generation

Column generation is often used to solve large scale optimization problems, and much research has been devoted to improve the convergence of the solution pr...

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A new algorithm, which exploits information from both ends of the network, is presented for the bicriterion shortest path problem. The algorithm extends eff...

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The disjoint bilinear programming problem can be reformulated using two distinct linear maxmin programming problems. There is a simple bijection between the...

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A new algorithm, based on interval analysis, is proposed for global optimization of constrained nonlinear nonconvex functions of several variables. It expl...

BibTeX reference**Pierre Hansen**

An algorithm based upon the boundary-edges code and the reverse search method is proposed for enumerating non-isomorphic planar simply connected polyhexes. ...

BibTeX reference**Pierre Hansen**and Nenad Mladenović

Systematic change of neighborhood within a possibly randomized local search algorithm yields a simple and effective metaheuristic for combinatorial and glo...

BibTeX referenceChopping Graphs

Chopping a graph <i>G</i> means removing one edge from each connected component of <i>G</i> in parallel, and repeating this until no edges remain. The requi...

BibTeX reference**Pierre Hansen**and Nenad Mladenović

Consider a set <i>L</i> of potential locations for <i>p</i> facilities and a set <i>U</i> of locations of given users. The <i>p</i>-median problem is to loc...

BibTeX reference**Pierre Hansen**

Finding extremal graphs for expressions involving one or more invariants is viewed as a problem of combinatorial optimization. The recent Variable Neighborh...

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We study the set of lower bounds which have been proposed for the numbering of a complete graph. We first show that the computation of most of them can be ...

BibTeX reference**Pierre Hansen**, Nenad Mladenović, and Éric Taillard

Good heuristic solutions for large Multisource Weber problems can be obtained by solving related <i>p</i>-median problems in which potential locations of th...

BibTeX referenceVariable Neighbourhood Search

**Pierre Hansen**

Systematic change of neighborhood within a local search algorithm yields a simple and effective metaheuristic for combinatorial optimization. We present a ...

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The multisource Weber problem is to locate simultaneously <i>m</i> facilities in the Euclidean plane in order to minimize the total transportation cost for ...

BibTeX referenceNesticity

**Pierre Hansen**and Dominique De Werra

A hypergraph <i>H=(X,E)</i> is <i>nested</i> if its edges are pairwise disjoint or included one into the other. The <i>nesticity</i> of <i>H</i> is defined...

BibTeX reference**Pierre Hansen**and Brigitte Jaumard

Given a set of entities, Cluster Analysis aims at finding subsets, called clusters, which are homogeneous and/or well separated. As many types of clusterin...

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La méthode de génération de colonnes est couramment utilisée pour résoudre des problèmes d'optimisation de grande taille. En pratique, on observe fréquemmen...

BibTeX reference**Pierre Hansen**, Panos M. Pardalos, and David J. Rader, Jr.

We study both the continuous and discrete problems of maximizing the product of two linear functions subject to all variables being between 0 and~1. We fir...

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It is shown that the anytime deduction procedure for probabilistic entailment of Frisch and Haddawy does not have the correctness property when the probabil...

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Clustering with a criterion which minimizes the sum of squared distances to cluster centroids is usually done in a heuristic way. An exact polynomial algori...

BibTeX reference**Pierre Hansen**, G Storchi, and T Vovor

Two new path problems in graphs are studied: MINRANGE, i.e., find a path from a vertex <i>s</i> to a vertex <i>t</i> with the smallest possible range of a...

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Two new algorithms are proposed for the problem of positioning a new product in attribute space in order to attract the maximum number of consumers. Each c...

BibTeX referenceProbabilistic Satisfiability

**Pierre Hansen**and Brigitte Jaumard

A review is made of models and algorithms for probabilistic satisfiability and its extensions. The basic probabilistic satisfiability problem, in decision f...

BibTeX reference**Pierre Hansen**and Nenad Mladenović

Extension of input-output functions for generating units allows simultaneous solution of the static unit commitment and economic dispatch problems by dynami...

BibTeX reference**Pierre Hansen**, Dominique Peeters, and Jacques-Francois Thisse

Zone pricing consists in determining simultaneously several delivered prices together with the zones where they are applied. A model and algorithm are prop...

BibTeX referenceMixed-Integer Column Generation Algorithms and the Probabilistic Maximum Satisfiability Problem

The column generation approach to large-scale linear programming is extended to the mixed-integer case. Two general algorithms, a dual and a primal one, are...

BibTeX reference**Pierre Hansen**, Fuji Zhang, and Maolin Zheng

We give lower and upper bounds for the number of reducible ears as well as upper bounds for the number of perfect matchings in an elementary bipartite graph...

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The maxmin problem models a game sequentially played by two players having opposite objective. Before making his move, the first player must anticipate the...

BibTeX referenceMixed Graph Colorings

**Pierre Hansen**, J. Kuplinsky, and Dominique De Werra

A mixed graph <i>G<sub><img src="theta.gif"></sub></i> contains both undirected edges and directed arcs. A <i>k</i>-coloring of <i>G<sub><img src="theta.g...

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D.-c. programming is a recent technique of global optimization, which allows the solution of problems whose objective function and constraints can be expres...

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We study links between the bilevel linear and linear mixed 0-1 programming problems. A new reformulation of the linear mixed 0-1 programming problem into a...

BibTeX reference**Pierre Hansen**and ID Moon

The <i>p</i>-maxisum dispersion problem consists of locating <i>p</i> facilities at vertices of a network in order to maximize the sum of the distances betw...

BibTeX reference**Pierre Hansen**, Martine Labbé, and M Minoux

A new location-allocation problem, called <i>p</i>-Center-Sum, is considered: given <i>n</i> clients (or demand points) locate <i>p</i> facilities among a g...

BibTeX referenceSplitting Trees

Splitting a tree is defined as removing all edges of a chain and disconnecting one from the other edges incident with that chain. Splitting a forest is simu...

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Weber's problem is to locate a facility in order to minimize the sum of its weighted distances to <i>n</i> fixed demand points. On a sphere, this problem i...

BibTeX reference**Pierre Hansen**and Brigitte Jaumard

Cluster Analysis is at the crossroad of many disciplines, and has numerous and diverse methods and applications. Mathematical Programming (together with gra...

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Consider an assignment problem in which persons are qualified for some but usually not all of the jobs. Moreover, assume persons belong to given seniority c...

BibTeX referenceGlobal Optimization in Location

Global optimization methods aim at finding the global optimum of a nonlinear and nonconvex function subject to nonlinear and nonconvex constraints. The main...

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An overview is given, with new results, of mathematical models and algorithms for probabilistic logic, probabilistic entailment and various extensions. Anal...

BibTeX reference**Pierre Hansen**, and Maolin Zheng

Cylindrical aromatic compounds - fullerenes and hydrocarbons - are about to become highly interesting objects of chemical research. A tubulene (tubular ben...

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Given a set of logical sentences together with probabilities that these sentences are true, the probabilistic satisfiability (PSAT) problem consists, in it...

BibTeX reference**Pierre Hansen**, DE Manolopoulos, and Maolin Zheng

A count of Kekulé structures for all 1812 distinct fullerene isomers of C<sub>60</sub> shows that 20 isomers surpass the count of 12500 for icosahedral C<su...

BibTeX referenceHow to Choose

*K*Entities Among

*N*

A new paradigm for cluster analysis is outlined. It is called Sequential Clustering. Given a set <i>O</i> of <i>N</i> entities a best cluster of <i>K</i> ...

BibTeX referenceRecursive and Explicit Formulae for the Degree of Freedom of Cata-Condensed Benzenoid Hydrocarbons

**Pierre Hansen**and Maolin Zheng

It is shown that the degree of freedom of a Kekulé structure <i>K</i> of a cata-condensed benzenoid hydrocarbon <i>H</i> is equal to the number of maximum d...

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Le problème de la satisfaisabilité probabiliste (PSAT) consiste à déterminer, étant donné les probabilités de <i>m</i> événements (ou propositions logiques...

BibTeX reference**Pierre Hansen**, C Lebatteux, and Maolin Zheng

The boundary-edges code for a benzenoid <i>H</i> is defined as the lexicographically maximum code obtained travelling around the boundary of <i>H</i> and no...

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An <i>O</i> (<i>mn</i> log <i>n</i>) labeling algorithm is proposed for minimum sum of diameters bipartitioning of the vertex set of a weighted graph (with ...

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Consider a set of logical sentences together with probabilities that they are true. These probabilities must satisfy certain conditions for this system to b...

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The product positioning problem consists in choosing the attributes of a new product in such a way as to maximize its market share, i.e., to attract a maxi...

BibTeX referenceLipschitz Optimization

**Pierre Hansen**and Brigitte Jaumard

Lipschitz optimization solves global optimization problems in which the objective function and constraint left-hand sides may be given by oracles (or explic...

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Much work has been devoted to the problem of finding maximum likelihood estimators for the three-parameter Weibull distribution. This problem has not been c...

BibTeX reference**Pierre Hansen**and Maolin Zheng

The Edmonds Matching Algorithm, which leads easily to finding a Kekulé structure in a chemical graph, is recalled. An extension is made to the case where on...

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The minimum sum-of-stars (or <i>p</i>-median, or <i>p</i>-medianoïd) clustering problem consists in partitioning a given set of entities into <i>P</i> clust...

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Three main mathematical programming approaches have been followed to design exact algorithms for partitioning problems in cluster analysis, cutting-planes, ...

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Dendrograms are widely used to represent graphically the clusters and partitions obtained with hierarchical clustering schemes. Espaliers are generalized de...

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We propose a new coding algorithm (called NTUPLE) for computing the N-tuple code (due to Knop et al.) for chemical trees. The algorithm NTUPLE has a ti...

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A linear algorithm is proposed for a particular case of the satisfiability problem which strictly includes nested satisfiability. Recognition of this case c...

BibTeX reference**Pierre Hansen**and Maolin Zheng

Consider a polyhex <i>H</i> which admits a perfect matching <i>M</i>. Harary, Klein and Zivkovic define the perfect matching vector of <i>M</i> as <img src...

BibTeX referenceBonds Fixed by Fixing Bounds

**Pierre Hansen**and Maolin Zheng

In a normal benzenoid hydrocarbon <i>H</i> no bonds are fixed, i.e. each bond belongs to some perfect matching or Kekulé structure of <i>H</i>. Fixing one o...

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We recently proposed to model location and sizing of offshore platforms for oil exploration and production as a multicapacitated plant location problem. We...

BibTeX reference**Pierre Hansen**and Brigitte Jaumard

Consider a set <i>O</i> of <i>N</i> entities and a matrix <i>D</i> = (<i>d<sub>k <img src="ell.gif"></sub></i>) of dissimilarities between pairs of entities...

BibTeX reference**Pierre Hansen**

Let <i>H</i> denote a simply-connected cata-condensed polyhex. It is shown that if <i>H</i> has three hexagons in a row it does not have a maximum number of...

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In a previous paper, a global optimization algorithm, called MLEW, is provided for finding Maximum Likelihood Estimators for the three-parameter Weibull dis...

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We consider nonlinear programs in 0-1 variables with nonlinear constraints and survey the main approaches to their solution: (i) linearization; (ii) algebra...

BibTeX reference**Pierre Hansen**and Maolin Zheng

A benzenoid system <i>H</i> is a finite connected subgraph of the infinite hexagonal lattice without cut bonds and non-hexagonal interior faces. The branchi...

BibTeX reference**Pierre Hansen**and Maolin Zheng

Let <i>N =</i> (<i>V,E</i>) be an undirected network with <i>n</i> vertices and <i>m</i> edges (i.e., |<i>V</i>| = <i>n</i> and |<i>E</i>| = <i>m</i>) in w...

BibTeX reference

An exact algorithm is proposed for average-linkage divisive hierarchical clustering, i.e., at each level of the hierarchy a cluster is bipartitioned in such...

BibTeX reference**Pierre Hansen**and Maolin Zheng

It is shown that the Clar number of a benzenoid hydrocarbon <i>H</i> (defined as the number of circles in a Clar formula, or equivalently as the maximum num...

BibTeX reference**Pierre Hansen**and Maolin Zheng

A proof is given that any normal benzenoid system with <i>h ></i> 1 hexagons has two hexagons the removal of each of which results in a normal benzenoid s...

BibTeX reference**Pierre Hansen**and Maolin Zheng

We consider graphs <i>G = </i>(<i>V, E</i>) with order <i>p</i> = |<i>V</i>|, size <i>e</i> = |<i>E</i>| and stability number <i><img src="beta.gif" align=m...

BibTeX reference**Pierre Hansen**and Nenad Mladenović

Five recent practically efficient methods for solving the maximum clique problem are briefly described and compared on randomly generated graphs. A Fortran ...

BibTeX reference**Pierre Hansen**and Fred S. Roberts

We show that certain reasonable axioms for an optimal solution to the problem of locating a facility on a network, i.e., axioms of distance determination, P...

BibTeX reference**Pierre Hansen**and Maolin Zheng

Many properties of benzenoid hydrocarbons can be explained in terms of their maximum number of mutually resonant hexagons. It is shown that this number can ...

BibTeX reference**Pierre Hansen**and Nenad Mladenović

It is shown that any vertex which is simplicial in the complementary graph of a given graph belongs to at least one maximum clique of that graph. This prope...

BibTeX reference**Pierre Hansen**and Maolin Zheng

Consider a benzenoid system with fixed bonds and the subgraph obtained by deleting fixed double bonds together with their end vertices and fixed single bond...

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The off-line vertex enumeration problem for polytopes consists in determining all vertices of a given polytope <i>P</i>. The on-line vertex enumeration prob...

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A recent global optimization algorithm using decomposition (GOP), due to Floudas and Visweswaran, when specialized to the case of polynomial functions is sh...

BibTeX reference**Pierre Hansen**and Maolin Zheng

A method to compute various upper bounds for the Clar number (defined as the number of circles in a Clar formula) of a benzenoid hydrocarbon is given. Based...

BibTeX reference

Weber's problem consists in locating a facility in the plane in such a way that the weighted sum of Euclidean distances to <i>n</i> given points be minimum....

BibTeX reference**Pierre Hansen**and Maolin Zheng

A hexagonal system is a connected plane graph without cut vertices in which each interior face is a regular hexagon. A linear algorithm is proposed to find ...

BibTeX referenceOn Clar Graphs

**Pierre Hansen**and Maolin Zheng

A necessary and sufficient condition is given for the Clar graph and the inner dual of a benzenoid system to be equal. It is shown that <i>k ></i> 2 hexag...

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A branch-and-bound algorithm is proposed for global minimization of indefinite quadratic functions subject to box constraints. Branching is done according ...

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Interval arithmetic and Taylor's formula can be used to bound the slope of the cord of a univariate function at a given point. Such bounds for the function,...

BibTeX reference**Pierre Hansen**and K-W Lih

A new heuristic algorithm, based on the Tabu Search methodology, is proposed for constrained redundancy optimization in series and in complex systems. It h...

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Konno and Inori (1989) have recently proposed two flexible models for bond portfolio optimization, together with heuristics to solve them. It is shown that ...

BibTeX reference**Pierre Hansen**and Maolin Zheng

A hexagonal system is a connected plane graph without cut vertices in which each interior face is a regular hexagon. A linear algorithm is proposed to find ...

BibTeX reference**Pierre Hansen**

We present a new class of change in risk, which includes several previous ones (price squeeze, strong increase in risk of Meyer and Ormiston) and leads to u...

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La programmation linéaire généralisée peut être étendue au cas des programmes mixtes en utilisant seulement l'algorithme primal révisé du simplexe en variab...

BibTeX reference**Pierre Hansen**and Brigitte Jaumard

Indefinite quadratic programs with quadratic constraints can be reduced to bilinear programs with bilinear constraints by duplication of variables. Such red...

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A new branch-and-bound algorithm for linear bilevel programming is proposed. Necessary optimality conditions expressed in terms of tightness of the follow...

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A bounded vertex coloring of a graph <i>G</i> is a usual vertex coloring in which each color is used at most <i>k</i> times (where <i>k</i> is a given numbe...

BibTeX reference**Pierre Hansen**and Maolin Zheng

It is proved that for any hexagonal system, there is a peak and a valley which are border adjacent (the path from the peak to the valley along the border is...

BibTeX referenceSlightly Hard-to-Color Graphs

**Pierre Hansen**and J. Kuplinsky

A graph is said to be slightly hard to color for a given vertex-coloring heuristic if some implementation of the algorithm uses more colors than are necessa...

BibTeX reference**Pierre Hansen**, Martine Labbé, and Jacques-Francois Thisse

The purpose of this paper is twofold. First, we revisit the cent-dian location problem developed by Halpern, considering both the average and maximum distan...

BibTeX reference**Pierre Hansen**

We propose an algorithm to compute the optimum departure time and path for a commuter in a congested network. Constant costs for use of arc, cost functions ...

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Nilsson recently introduced a rigorous semantic generalization of logic in which the truth values of sentences are probability values. This led to state pre...

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The problem of optimal location and sizing of off-shore platforms for oil exploration can be formulated as follows: given a set of oil wells to be drilled a...

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Timonov proposes an algorithm for global maximization of univariate Lipschitz functions in which successive evaluation points are chosen in order to ensure ...

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Divisive hierarchical clustering algorithms with the diametercriterion proceed by recursively selecting the cluster with largest diameter and partitioning i...

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We study the complexity and propose an algorithm for the problem of determining, given <i>p</i> vectors of {-1, 1}<i><sup>n</sup></i>, all linear combinatio...

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Consider <i>N</i> entities to be classified, with given weights, and a matrix of dissimilarities between pairs of them. The split of a cluster is the small...

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Unconstrained hyperbolic 0-1 programming can be solved in linear time when the numerator and the denominator are linear and the latter is always positive. I...

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Several authors have proposed to estimate Lipschitz constants in global optimization by a multiple of the largest slope (in absolute value) between successi...

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An "intelligent front-end" or "logic assistant" is an interactive program devised to assist the users of an information retrieval system in the formulation ...

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We propose a new algorithm to solve the on-line vertex enumeration problem for polytopes, doing all computations in <i>n</i>-space, where <i>n</i> is the di...

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We consider the following global optimization problems for a Lipschitz function <i>f</i> implicitly defined on an interval [<i>a,b</i>]. Problem <i>P'</i>:...

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We consider the following global optimization problems for a univariate Lipschitz function <i>f</i> defined on an interval [<i>a,b</i>]: Problem <i>P</i>: f...

BibTeX referenceMaximum Sum-of-Splits Clustering

FORTRAN code of an efficient implementation of a <img src="Theta.gif" align=bottom> (<i>N</i><sub>2</sub>) algorithm for the maximum sum-of-splits clusterin...

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Global optimization problems with a few variables and constraints arise in numerous applications but are seldom solved exactly. Most often only a local opti...

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Piyavskii's algorithm maximizes a univariate function <i>f</i> satisfying a Lipschitz condition. We compare the numbers of iterations needed to obtain a bou...

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Consider <i>N</i> entities to be classified, and a matrix of dissimilarities between pairs of them. The split of a cluster is the smallest dissimilarity be...

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The Basic Algorithm of pseudo-Boolean programming due to Hammer and Rudeanu allows to minimize nonlinear 0-1 functions by recursively eliminating one variabl...

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An experimental computer system for globally optimal design, called BAGOP, is discussed. This new tool uses the computer alebra system MACSYMA to implement ...

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Many problems of globally optimal design have been solved in the literature using monotonicity analysis and a variety of tests applied in an ad hoc way. The...

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Many problems of globally optimal design have been solved by using monotonicity analysis. New monotonicity principles are obtained by exploiting non strict ...

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Ordered sequential algorithms for the global minimization of univariate functions over an interval proceed by evaluating this function at successive points c...

BibTeX referenceMaximum Sum of Splits Clustering

**Pierre Hansen**and Brigitte Jaumard

Consider N entities to be classified, and a matrix of diffimilarities between pairs of them. The split of a cluster is the smallest dissimilarity between an...

BibTeX reference**Pierre Hansen**and Brigitte Jaumard

Old and new algorithms for the Maximum Satisfiability problem are studied. We first summarize the different heuristics previously proposed, i.e. the approxi...

BibTeX reference### Articles

**Pierre Hansen**, and Frédéric Messine

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**, and Dragutin Svrtan

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**, Mustapha Aouchiche, Gilles Caporossi, Alain Hertz, and Cherif Sellal

**Pierre Hansen**

**Pierre Hansen**, Alain Hertz, Cherif Sellal, Damir Vukicevic, Mustapha Aouchiche, and Gilles Caporossi

**Pierre Hansen**, Nenad Mladenović, Raca Todosijević, and Saïd Hanafi

**Pierre Hansen**

**Pierre Hansen**, and Nenad Mladenović

**Pierre Hansen**

**Pierre Hansen**, Bassem Jarboui, and Nenad Mladenović

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**, and Frédéric Messine

**Pierre Hansen**, and Nenad Mladenović

**Pierre Hansen**

**Pierre Hansen**, Rita Macedo, and Nenad Mladenović

**Pierre Hansen**, and Leo Liberti

**Pierre Hansen**, and Sylvain Perron

**Pierre Hansen**, and Gilbert Laporte

**Pierre Hansen**

**Pierre Hansen**, Caroline Rocha, and Éverton Santi

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**, and Nenad Mladenović

**Pierre Hansen**

**Pierre Hansen**, and Leo Liberti

**Pierre Hansen**, and Alain Hertz

**Pierre Hansen**, and Leo Liberti

**Pierre Hansen**, V. Maniezzo, and Stefan Voß

**Pierre Hansen**

**Pierre Hansen**, and Jack Brimberg

**Pierre Hansen**

**Pierre Hansen**, Frédéric Messine, and Jordan Ninin

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**, and Leo Liberti

**Pierre Hansen**, Lucas Létocart, Leo Liberti, and Frédéric Messine

**Pierre Hansen**, Manuel Ruiz, and Daniel Aloise

**Pierre Hansen**, and Leo Liberti

**Pierre Hansen**, and Leo Liberti

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**, Sylvain Perron, and Alberto Costa

**Pierre Hansen**, Frédéric Messine, and Sylvain Perron

**Pierre Hansen**, and Nenad Mladenović

**Pierre Hansen**and Christophe Meyer

**Pierre Hansen**

**Pierre Hansen**, and Sylvain Perron

**Pierre Hansen**

**Pierre Hansen**, and Alain Hertz

**Pierre Hansen**

**Pierre Hansen**, and Claire Lucas

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**, and F. Maffray

**Pierre Hansen**, Nenad Mladenović, and Eric W.T. Ngai

**Pierre Hansen**, and Leo Liberti

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**, and Frédéric Messine

**Pierre Hansen**

**Pierre Hansen**, and Nenad Mladenović

**Pierre Hansen**, Sébastien Le Digabel, and Nenad Mladenović

**Pierre Hansen**, and Dragan Stevanovic

**Pierre Hansen**and Claire Lucas

**Pierre Hansen**, and Damir Vukicevic

**Pierre Hansen**, and Leo Liberti

**Pierre Hansen**, and Leo Liberti

**Pierre Hansen**

**Pierre Hansen**, Leo Liberti, and Sylvain Perron

**Pierre Hansen**

**Pierre Hansen**, Nenad Mladenović, and J. Perez

**Pierre Hansen**, and P Popat

**Pierre Hansen**and Claire Lucas

**Pierre Hansen**, and Dragan Stevanovic

**Pierre Hansen**and Christophe Meyer

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**, Alain Hertz, Rim Kilani, Odile Marcotte, and David Schindl

**Pierre Hansen**, and Frédéric Messine

**Pierre Hansen**, and Frédéric Messine

**Pierre Hansen**

**Pierre Hansen**, and Frédéric Messine

**Pierre Hansen**

**Pierre Hansen**, Martine Labbé, and David Schindl

**Pierre Hansen**, and Frédéric Messine

**Pierre Hansen**, Jack Brimberg, Dragan Urosević, and Nenad Mladenović

**Pierre Hansen**, V. Maniezzo, and S. Voss

**Pierre Hansen**

**Pierre Hansen**and Damir Vukicevic

**Pierre Hansen**, and Frédéric Messine

**Pierre Hansen**

**Pierre Hansen**, C Oguz, and Nenad Mladenović

**Pierre Hansen**and Sylvain Perron

**Pierre Hansen**, Peter Rowlinson, S Simic, and Dragan Stevanovic

**Pierre Hansen**, Nenad Mladenović, and Said Salhi

**Pierre Hansen**and Nenad Mladenović

**Pierre Hansen**, Alejandro Karam, and Sylvain Perron

**Pierre Hansen**and Dragan Stevanovic

**Pierre Hansen**

**Pierre Hansen**, Gilbert Laporte, Nenad Mladenović, and Dragan Urosević

**Pierre Hansen**

**Pierre Hansen**, Nenad Mladenović, and José A. Moreno Pérez

**Pierre Hansen**

**Pierre Hansen**, and José A. Moreno Pérez

**Pierre Hansen**, and Frédéric Messine

**Pierre Hansen**

**Pierre Hansen**, Jasmina Lazic, and Nenad Mladenović

**Pierre Hansen**, Jack Brimberg, Dragan Urosević, and Nenad Mladenović

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**, and Frédéric Messine

**Pierre Hansen**, and Maolin Zheng

**Pierre Hansen**and Sylvain Perron

**Pierre Hansen**, Jean-Louis Lagouanelle, and Frédéric Messine

**Pierre Hansen**and Damir Vukicevic

**Pierre Hansen**

**Pierre Hansen**and Nenad Mladenović

**Pierre Hansen**

**Pierre Hansen**, Nenad Mladenović, and Dragan Urosević

**Pierre Hansen**, and Nenad Mladenović

**Pierre Hansen**, and Dragan Stevanovic

**Pierre Hansen**, and Maolin Zheng

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**, and Frédéric Messine

**Pierre Hansen**, Hadrien Mélot, and Ivan Gutman

**Pierre Hansen**, Mustapha Aouchiche, Gilles Caporossi, Hadrien Mélot, and Dragan Stevanovic

**Pierre Hansen**, Eric W.T. Ngai, Bernard K.-S. Cheung, and Nenad Mladenović

**Pierre Hansen**, and Nenad Mladenović

**Pierre Hansen**, and Hadrien Mélot

**Pierre Hansen**, and M Laffay

**Pierre Hansen**

**Pierre Hansen**, and Nenad Mladenović

**Pierre Hansen**, Sébastien Le Digabel, and Nenad Mladenović

**Pierre Hansen**

**Pierre Hansen**, Frédéric Messine, and Sylvain Perron

**Pierre Hansen**

**Pierre Hansen**, Nenad Mladenović, and Dragan Urosević

**Pierre Hansen**, and Vera Kovacevic-Vujcic

**Pierre Hansen**

**Pierre Hansen**, and Frédéric Messine

**Pierre Hansen**, and Nenad Mladenović

**Pierre Hansen**

**Pierre Hansen**, Brigitte Jaumard, Christophe Meyer, B Simeone, and V Doring

**Pierre Hansen**, K-W Lih, Nenad Mladenović, and Michèle Breton

**Pierre Hansen**, and L Pavlovic

**Pierre Hansen**and Hadrien Mélot

**Pierre Hansen**

**Pierre Hansen**, and Dragan Stevanovic

**Pierre Hansen**, and Federico Malucelli

**Pierre Hansen**, and Martine Labbé

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**and Nenad Mladenović

**Pierre Hansen**and Nenad Mladenović

**Pierre Hansen**and Nenad Mladenović

**Pierre Hansen**and Nenad Mladenović

**Pierre Hansen**

**Pierre Hansen**

### Books

**Pierre Hansen**

**Pierre Hansen**, and F. Maffray

**Pierre Hansen**

**Pierre Hansen**, and Gilles Savard

**Pierre Hansen**, and Fred S. Roberts

### Book chapters

**Pierre Hansen**, Nenad Mladenović, Jack Brimberg, and José A. Moreno Pérez

**Pierre Hansen**and Nenad Mladenović

**Pierre Hansen**, and Nenad Mladenović

**Pierre Hansen**, and Nenad Mladenović

**Pierre Hansen**, and Nenad Mladenović

**Pierre Hansen**and Christophe Meyer

**Pierre Hansen**

**Pierre Hansen**, Leo Liberti, Sylvain Perron, and Manuel Ruiz

**Pierre Hansen**, and Leo Liberti

**Pierre Hansen**, and Leo Liberti

**Pierre Hansen**, Leo Liberti, and Dario J. Aloise

**Pierre Hansen**, and Leo Liberti

**Pierre Hansen**, Nenad Mladenović, Jack Brimberg, and José A. Moreno Pérez

**Pierre Hansen**and Nenad Mladenović

**Pierre Hansen**, and Abdelwaheb Rebaï

**Pierre Hansen**, and Frédéric Messine

**Pierre Hansen**

**Pierre Hansen**, L Hiesse, J Lacheré, and A Monhait

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**, and Carla Silva Oliveira

**Pierre Hansen**and Hadrien Mélot

**Pierre Hansen**, Nenad Mladenović, Jack Brimberg, and José A. Moreno Pérez

**Pierre Hansen**, and Nenad Mladenović

**Pierre Hansen**, and Sébastien Le Digabel

### Proceedings

**Pierre Hansen**, Aidan Riordan, and Xavier Hansen

**Pierre Hansen**

**Pierre Hansen**, and Frédéric Messine

**Pierre Hansen**

**Pierre Hansen**, Nenad Mladenović, and Claire Lucas

**Pierre Hansen**

**Pierre Hansen**, and Sylvain Perron

**Pierre Hansen**, and Caroline Rocha

**Pierre Hansen**

**Pierre Hansen**, and Caroline Rocha

**Pierre Hansen**

**Pierre Hansen**

**Pierre Hansen**, Leo Liberti, Sylvain Perron, and Manuel Ruiz

**Pierre Hansen**

**Pierre Hansen**, and Leo Liberti

**Pierre Hansen**

**Pierre Hansen**, and Leo Liberti

**Pierre Hansen**, and Leo Liberti

**Pierre Hansen**, and Leo Liberti

**Pierre Hansen**

**Pierre Hansen**, Yuri Kochetov, and Nenad Mladenović