Journées de l'optimisation 2007 Optimizations days

We propose to apply the Parallel Variable Distribution (PVD) technique from Ferris and Magasarian to the Mesh Adaptive Direct Search (MADS) algorithm, which extends the Generalized Pattern Search (GPS) algorithm for nonsmooth optimization. The resulting algorithm intends to solve large problems by using parallel MADS instances optimizing only a portion of the variables. Numerical results illustrate advantages and limitations of this method.

The present work deals with a portfolio selection in the presence of skewness. Within this framework, we propose a new approach to solve the multi-objective portfolio problem. Several researchers have proved the existence of a risk premium for skewness. Nevertheless, only a few studies incorporate the skewness in the selection of portfolio. In the context of skewness, the selection of efficient portfolio requires the optimization of different and conflicting criteria such as maximizing expected return and skewness and minimizing risk. Hence, the portfolio selection can be formulated as a tri-objective programming problem to solve the mean-variance-skewness efficient set. Lai (1991) suggests that the portfolio choice can be rescaled on the unit variance space. This suggestion reduces the number of objective functions to two, but adds an equality constraint to the model. Since non-linear equality constraints are difficult to handle in optimization framework, solutions obtained by Lai (1991) do not exactly satisfy the constraint : the variance of portfolios is not exactly equal to one. Then, Lai (1991) transforms the resulting multi-objective problem into a polynomial goal programming problem PGP by incorporating the investor a priori preference between objectives. Modeling the decision maker preferences could be difficult and skewed. We propose a new approach to generate the entire efficient set. The approach is independent to the relative and subjective preferences of investor. Having the efficient set, the investor decides a posteriori the solution which is appropriate to him. First, the model presented by Lai (1991) is reformulated to an equivalent unconstrained problem by changing variables and exploiting the properties of variance matrix. This reformulation simplifies the resolution of the problem while keeping the feasibility of the solutions. Then, a new algorithm BIMADS is applied to generate an efficient set of portfolios. The approach is tested on the example presented by Lai (1991). In this example, five securities and a risk-free asset are used. The historical data (provided by the Wall Street Journal and CRSP) are used to estimate expected return, covariance and central comment. Our approach generates a set of efficient portfolios. More than 1500 efficient solutions are generated by applying one BIMADS while the PGP algorithm proposed by Lai (1991) generates 6 solutions with 6 different combinations of preferences. As opposed to the solutions proposed by Lai (1991), the ones generated by our new approach are feasible since they satisfy the condition of unit variance.

Fitting a reliability model is of great importance in system safety. Forecast of degradation and finding an optimal maintenance schedule are the result of several sub-problem optimizations. One of them is to fit a probability density function to field degradation data. We show how to use a non smooth optimization algorithm to estimate four parameters of a probability density function compounded of two stochastic processes. This distribution is computed numerically and parameters of the model are evaluated in the case of fatigue crack growth using maximum likelihood function.

In the framework of linear programming, we propose a theoretical justification for regularizing the linear systems used to compute search directions when the latter are (nearly) rank deficient. We present a primal-dual infeasible algorithm for linear programs (LPs) with explicit primal and dual regularization. We establish a connection between proximal-point, regularization, trust-region, and augmented Lagrangian methods. The regularization is termed \emph{exact} to emphasize that, although the LP is regularized, we are able to recover a solution of the original LP, independently of the values of the regularization parameters.