Weak Structural Dependence in Chance-Constrained Programming

In chance-constrained optimization problems, a solution is assumed to be feasible only with certain, sufficiently high probability. For computational and theoretical purposes, the convexity property of the resulting constraint set is treated. It is known, for example, that a suitable combination of a concavity property of the probability distribution and concavity of constraint mappings are sufficient conditions to the convexity of the resulting constraint set. Recently, new concavity condition of the probability distribution r-decreasing density has been developed. Henrion and Strugarek (2006) show, under the assumption of independence of constraint rows, that this condition on marginal densities allows us, on the other side, weaken the concavity of constraint mappings. In this contribution we present a relaxation of the independence assumption in favour of a specific weak-dependence condition. If the independence assumption is not fulfiled, the resulting constraint set is not due to be convex. However, under a weak-dependence assumption, the non-convex problem can be approximated by a convex one. Applying stability results on optimal values and optimal solutions, we show that optimal values and optimal solutions remain stable under assumptions common in stochastic programming. This implies desirable consequences, because convex problems are easiest to compute and also many theoretical results are based on convexity assumptions. We accompany the shown results by simple example to illustrate the concept of the presented approximation.