多阶段有补偿问题的非线性模型

通过对已有补偿问题的模型进行总结，抽象与升华，本文建立了Ｂａｎａｃｈ空间中一般形式多阶段有补偿随机规划问题的一个非线性模型，使已有所有的补偿问题均成为其特例；然后利用可测集值映射理论，正规凸的被积函数的性质及文［８」中的结论等，讨论了所给模型的适定性与其基本性质．     

The empirical behavior of sampling methods for stochastic programming

We investigate the quality of solutions obtained from sample-average approximations to two-stage stochastic linear programs with recourse. We use a recently developed software tool executing on a computational grid to solve many large instances of these problems, allowing us to obtain high-quality solutions and to verify optimality and near-optimality of the computed solutions in various ways. Research supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, U.S. Department of Energy, under Contract W-31-109-Eng-38, and by the National Science Foundation under Grant 9726385. Research supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, U.S. Department of Energy, under Contract W-31-109-Eng-38, and by the National Science Foundation under Grant DMS-0073770. Research supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, U.S. Department of Energy, under Contract W-31-109-Eng-38, and by the National Science Foundation under Grants 9726385 and 0082065.     

Piecewise convex programs

A piecewise convex program is a convex program such that the constraint set can be decomposed in a finite number of closed convex sets, called the cells of the decomposition, and such that on each of these cells the objective function can be described by a continuously differentiable convex function.In a first part, a cutting hyperplane method is proposed, which successively considers the various cells of the decomposition, checks whether the cell contains an optimal solution to the problem, and, if not, imposes a convexity cut which rejects the whole cell from the feasibility region. This elimination, which is basically a dual decomposition method but with an efficient use of the specific structure of the problem is shown to be finitely convergent.The second part of this paper is devoted to the study of some special cases of piecewise convex program and in particular the piecewise quadratic program having a polyhedral constraint set. Such a program arises naturally in stochastic quadratic programming with recourse, which is the subject of the last section.This paper is based on the author's Ph.D. Dissertation presented at the Faculté des Sciences Appliquées of the Université Catholique de Louvain. It describes research supported partly by the Programme National d'Impulsion à la Recherche en Informatique of the Belgian Government under contract No. I (14 bis) 6 and partly by a two-year fellowship of the Centre Interuniversitaire d'Etudes Doctorales dans les Sciences du Management.     

A priori orienteering with time windows and stochastic wait times at customers

In the pharmaceutical industry, sales representatives visit doctors to inform them of their products and encourage them to become an active prescriber. On a daily basis, pharmaceutical sales representatives must decide which doctors to visit and the order to visit them. This situation motivates a problem we more generally refer to as a stochastic orienteering problem with time windows (SOPTW), in which a time window is associated with each customer and an uncertain wait time at a customer results from a queue of competing sales representatives. We develop a priori routes with the objective of maximizing expected sales. We operationalize the sales representative’s execution of the a priori route with relevant recourse actions and derive an analytical formula to compute the expected sales from an a priori tour. We tailor a variable neighborhood search heuristic to solve the problem. We demonstrate the value of modeling uncertainty by comparing the solutions to our model to solutions of a deterministic version using expected values of the associated random variables. We also compute an empirical upper bound on our solutions by solving deterministic instances corresponding to perfect information.     

Stochastic semidefinite programming: a new paradigm for stochastic optimization

Semidefinite programs are a class of optimization problems that have been studied extensively during the past 15 years. Semidefinite programs are naturally related to linear programs, and both are defined using deterministic data. Stochastic programs were introduced in the 1950s as a paradigm for dealing with uncertainty in data defining linear programs. In this paper, we introduce stochastic semidefinite programs as a paradigm for dealing with uncertainty in data defining semidefinite programs.The work of this author was supported in part by the U.S. Army Research Office under Grant DAAD 19-00-1-0465. The material in this paper is part of the doctoral dissertation of this author in preparation at Washington State University.     

A tighter variant of Jensen’s lower bound for stochastic programs and separable approximations to recourse functions

In this paper, we propose a new method to compute lower bounds on the optimal objective value of a stochastic program and show how this method can be used to construct separable approximations to the recourse functions. We show that our method yields tighter lower bounds than Jensen’s lower bound and it requires a reasonable amount of computational effort even for large problems. The fundamental idea behind our method is to relax certain constraints by associating dual multipliers with them. This yields a smaller stochastic program that is easier to solve. We particularly focus on the special case where we relax all but one of the constraints. In this case, the recourse functions of the smaller stochastic program are one dimensional functions. We use these one dimensional recourse functions to construct separable approximations to the original recourse functions. Computational experiments indicate that our lower bounds can significantly improve Jensen’s lower bound and our recourse function approximations can provide good solutions.     

Stability in stochastic programming with recourse-estimated parameters

In this paper, stability of the optimal solution of stochastic programs with recourse with respect to parameters of the given distribution of random coefficients is studied. Provided that the set of admissible solutions is defined by equality constraints only, asymptotical normality of the optimal solution follows by standard methods. If nonnegativity constraints are taken into account the problem is solved under assumption of strict complementarity known from the theory of nonlinear programming (Theorem 1). The general results are applied to the simple recourse problem with random right-hand sides under various assumptions on the underlying distribution (Theorems 2–4).