Stochastic Two-Stage Programming

by Karl Frauendorfer

Overview

Shastic Programming offers models and methods for decision problems wheresome of the data are uncertain. These models have features and structural properties which are preferably exploited by SP methods within the solution process. This work contributes to the methodology for two-stagemodels. In these models the objective function is given as an integral, whose integrand depends on a random vector, on its probability measure and on a decision. The main results of this work have been derived with the intention to ease these difficulties: After investigating duality relations for convex optimization problems with supply/demand and prices being treated as parameters, a stability criterion is stated and proves subdifferentiability of the value function. This criterion is employed for proving the existence of bilinear functions, which minorize/majorize the integrand. Additionally, these minorants/majorants support the integrand on generalized barycenters of simplicial faces of specially shaped polytopes and amount to an approach which is denoted barycentric approximation scheme.

Product Details

ISBN-13:	9783540560975
Publisher:	Springer Berlin Heidelberg
Publication date:	12/08/1992
Series:	Lecture Notes in Economics and Mathematical Systems , #392
Edition description:	Softcover reprint of the original 1st ed. 1992
Pages:	228
Product dimensions:	6.69(w) x 9.53(h) x 0.02(d)

Table of Contents

0 Preliminaries.- I Shastic Two-Stage Problems.- 1 Convex Case.- 2 Nonconvex Case.- 3 Stability.- 4 Epi-Convergence.- 5 Saddle Property.- 6 Shastic Independence.- 7 Special Convex Cases.- II Duality and Stability in Convex Optimization (Extended Results for the Saddle Case).- 8 Characterization and Properties of Saddle Functions.- 9 Primal and Dual Collections of Programs.- 10 Normal and Stable Programs.- 11 Relation to McLinden’s Results.- 12 Application to Convex Programming.- III Barycentric Approximation.- 13 Inequalities and Extremal Probability Measures — Convex Case.- 14 Inequalities and Extremal Probability Measures — Saddle Case.- 15 Examples and Geometric Interpretation.- 16 Iterated Approximation and x-Simplicial Refinement.- 17 Application to Shastic Two-Stage Programs.- 18 Convergence of Approximations.- 19 Refinement Strategy.- 20 Iterative Completion.- IV An Illustrative Survey of Existing Approaches in Shastic Two-Stage Programming.- 21 Error Bounds for Shastic Programs with Recourse (due to Kali & Stoyan).- 22 Approximation Schemes discussed by Birge & Wets.- 23 Sublinear Bounding Technique (due to Birge & Wets).- 24 Shastic Quasigradient Techniques (due to Ermoliev).- 25 Semi-Shastic Approximation (due to Marti).- 26 Benders’ Decomposition with Importance Sampling (due to Dantzig & Glynn).- 27 Shastic Decomposition (due to Higle & Sen).- 28 Mathematical Programming Techniques.- 29 Scenarios and Policy Aggregation (due to Rockafellar & Wets).- V BRAIN — BaRycentric Approximation for Integrands (Implementation Issues).- 30 Storing Distributions given through a Finite Set of Parameters.- 31 Evaluation of Initial Extremal Marginal Distributions.- 32 Evaluation of Initial Outer and Inner Approximation.- 33 Data forx-Simplicial Partition.- 34 Evaluation of Extremal Distributions — Iteration!.- 35 Evaluation of Outer and Inner Approximation — Iteration J.- 36 x-Simplicial Refinement.- VI Solving Shastic Linear Two-Stage Problems (Numerical Results and Computational Experiences).- 37 Testproblems from Literature.- 38 Randomly Generated Testproblems.