Conditional Value-at-Risk in Stochastic Programs with Mixed-Integer Recourse

Abstract

In classical two-stage stochastic programming the expected value of the total costs is minimized. Recently, mean-risk models-studied in mathematical finance for several decades-have attracted attention in stochastic programming. We consider Conditional Value-at-Risk as risk measure in the framework of two-stage stochastic integer programming. The paper addresses structure, stability, and algorithms for this class of models. In particular, we study continuity properties of the objective function, both with respect to the first-stage decisions and the integrating probability measure. Further, we present an explicit mixed-integer linear programming formulation of the problem when the probability distribution is discrete and finite. Finally, a solution algorithm based on Lagrangean relaxation of nonanticipativity is proposed.

Shape Optimization under Uncertainty from a Stochastic Programming Point of View

“…Of course, the actual definition of risk depends on the decision maker and is rather arbitrary, which is why we only focus on two risk measures here that have been studied in the context of two-stage stochastic programming, namely Expected Excess and Excess Probability. There are other risk measures such as the Value-at-Risk or the Conditional Value-at-Risk commonly used in mathematical finance, and we only refer the reader to [81,82,93] for details on Introduction those. We introduce the two risk measures we are concerned with in this thesis in the following definition.…”

“…Schultz and Tiedemann showed in [82,Lemma 4.4] that it is possible to rewrite the expected excess Q EEη in the form of Q E with a suitably adapted second-stage program which satisfies the required assumptions. However, their models additionally include integer requirements on some of the variables whereas ours do not.…”

“…Yet it is easy to adapt the quoted Lemma to our situation, and it will turn out to be useful later in Section 3.3 by providing a way to solve the expected excess shape optimization problem numerically. The proof is based on the one presented in [82]. Lemma 1.13.…”

“…In some applications, there are additionally integer requirements some or all components of x or y have to comply with. We are not concerned with any integer constraints here but refer the reader to [60,79,80,82,93] for completeness. When looking at the formulation (1.20), it immediately becomes apparent that this very formulation does actually not make much sense, at least if one tries to find a solution.…”

Convex approximations for a class of mixed-integer recourse models

“…A branch-andprice algorithm is proposed in Silva and Wood (2006) to solve a stochastic facility-location problem. Finally, (Schultz and Tiedemann 2003;Schultz and Tiedemann 2006) consider mean-risk versions of mixed-integer recourse problems.…”