Groupe d’études et de recherche en analyse des décisions

G-2015-03

Conic optimization

Conic optimization refers to the problem of optimizing a linear function over the intersection of an affine space and a closed convex cone. Conic optimization problems are thus a particular class of convex optimization problems. We focus particularly on the special case where the cone is chosen as the cone of positive semidefinite matrices for which the resulting optimization problem is called a semidefinite optimization problem. The class of semidefinite optimization problems includes linear optimization problems as a special case, namely when all the matrices involved are diagonal. Another special case of semidefinite optimization is second-order cone optimization that corresponds to optimizing over the second-order cone, also known as the Lorentz cone. As the case of linear optimization is discussed in detail in elsewhere in this book, we focus here on the use of other cones. Although most research has focused on the positive semidefinite cone, the second-order cone is arguably more important in practical applications, as shown for example in the chapter on Financial Enginering in this part.

This chapter provides an introduction to conic optimization, and the three subsequent chapters present applications of conic optimization in control engineering, truss topology design, and financial engineering. The reader wishing to delve deeper into the area is refereed to the books

  • H. Wolkowicz, R. Saigal, and L. Vandenberghe, editors. Handbook of Semidenite Programming. Kluwer Academic Publishers, Boston, MA, 2000.
  • S. Boyd and L. Vandenberghe. Convex optimization. Cambridge University Press, Cambridge, 2004.
  • M.F. Anjos and J.B. Lasserre, editors. Handbook on Semidenite, Conic and Polynomial Optimization. International Series in Operations Research & Management Science. Springer-Verlag, 2011.

and their extensive bibliographies.

, 13 pages