The two-stage stochastic linear program, today a classic, has seen various extensions in the recent and not so recent past. We will discuss three of them in the talk. Mixed-integer recourse models, clearly indispensable in many applications, bear the difficulty that convexity of the objective, which is of paramount importance for all amenities met with the classical linear recourse model, is lost. Structural analysis and algorithmic treatment must be rethought from the very beginning. The results are more discontinuities in the structures and algorithms, which are closer cousins to the basic methods in integer programming rather than convex optimization. Dominance constraints induced by linear or mixed-integer linear recourse are flexible tools for handling risk aversion, a feature not covered by the classical two-stage model. Structurally, investigations are directed to the constraints rather the objective. Algorithmically, mixed integer linear programming equivalents are much bigger and tighter coupled internally than the classical models. PDE constraints were studied only very recently in two-stage stochastic programming. Shape optimization of bodies under linear elasticity and stochastic loading serves as an example to demonstrate how the basic paradigm of two-stage stochastic programming straightforwardly extends: In the first-stage, decide on the shape so that the random forces attacking thereafter, in the mean or according to risk measure, produce the least compliance of the body. Structurally, this leads to the study of shape gradients, i.e., calculus on spaces whose elements are open sets. Algorithmically, it takes advanced nite-element methods and algorithmic shortcuts avoiding repeated solution of PDEs for the complete list of load scenarios.
[Gainesville: FL] Dr. Rudiger Schultz, Department of Mathematics, University of Duisburg-Essen, Duisburg Germany will deliver a seminar entitled, “Three Classes of Two-Stage Stochastic Programs: Mixed-Integer Recourse, Dominance, and PDE constraints” on Thursday, March 29 from 3:00 to 3:50 p.m. in room 279, Weil Hall.