Roger J.-B. Wets

Minimization by Random Search Techniques

We give two general convergence proofs for random search algorithms. We review the literature and show how our results extend those available for specific variants of the conceptual algorithm studied here. We then exploit the convergence results to examine convergence rates and to actually design implementable methods. Finally we report on some computational experience.

L-SHAPED LINEAR PROGRAMS WITH APPLICATIONS TO OPTIMAL CONTROL AND STOCHASTIC PROGRAMMING.

This paper gives an algorithm for L-shaped linear programs which arise naturally in optimal control problems with state constraints and stochastic linear programs (which can be represented in this form with an infinite number of linear constraints). The first section describes a cutting hyperplane algorithm which is shown to be equivalent to a partial decomposition algorithm of the dual program. The two last sections are devoted to applications of the cutting hyperplane algorithm to a linear optimal control problem and stochastic programming problems.

Stochastic Programming: Solution Techniques and Approximation Schemes

Solutions techniques for stochastic programs are reviewed. Particular emphasis is placed on those methods that allow us to proceed by approximation. We consider both stochastic programs with recourse and stochastic programs with chance-constraints.

Programming Under Uncertainty: The Equivalent Convex Program

This paper is an attempt to describe and characterize the equivalent convex program of a two-stage linear program under uncertainty. The study has been divided into two parts. In the first one, we examine the properties of the solution set of the problem and derive explicit expressions for some particular cases. The second section is devoted to the derivation of the objective function of the equivalent convex program. We show that it is convex and continuous. We also give a necessary condition for its differentiability and establish necessary and sufficient conditions for the solvability of the problem. Finally, we give the equivalent convex program of certain classes of programming under uncertainty problems, i.e., when the constraints and the probability space have particular structures.

Quantitative stability of variational systems. I.\ The epigraphical distance

This paper proposes a global measure for the distance between the elements of a variational system (parametrized families of optimization problems).

Solving stochastic programss with simple recourse

In this paper we describe an algorithm for solving stochastic programs with simplerecourse, i.e.,generated by a linear programming problem with stochastic coefficients and a specific loss function ...

Epi‐consistency of convex stochastic programs

This paper presents consistency results for sequences of optimal solutions to convex stochastic optimization problems constructed from empirical data, by applying the strong law of large numbers fo...

On decision rules in stochastic programming

The paper surveys the basic results and nonresults for decision rules in stochastic programming. It exhibits some of the difficulties encountered when trying to restrict the class of acceptable rules to those possessing specific functional forms. A liberal dosage of examples is provided which illustrate various cases. The treatment is unified by making use of the equivalence of various formulations which have appeared in the literature. An appendix is devoted to the P-model for stochastic programs with chance constraints.

Stability in two-stage stochastic programming

We analyze the effect of changes in problem functions and/or distributions in certain two-stage stochastic programming problems with recourse. Under reasonable assumptions the locally optimal value of the perturbed problem will be continuous and the corresponding set of local optimizers will be upper semicontinuous with respect to the parameters (including the probability distribution in the second stage).

A class of stochastic programs withdecision dependent random elements

In the “standard” formulation of a stochastic program with recourse, the distribution ofthe random parameters is independent of the decisions. When this is not the case, the problemis significantly more difficult to solve. This paper identifies a class of problems that are“manageable” and proposes an algorithmic procedure for solving problems of this type. Wegive bounds and algorithms for the case where the distributions and the variables controllinginformation discovery are discrete. Computational experience is reported.