Optimization Models with Probabilistic Constraints

Summary

This chapter presents an overview of the theory and numerical techniques for optimization models involving one or more constraints on probability functions. We focus on recent developments involving nonlinear probabilistic models. The theoretical fundament includes the theory and examples of generalized concavity for functions and measures, and some specific properties of probability distributions, including discrete distributions. We analyze the structure and properties of the constraining probabilistic functions and of the probabilistically constrained sets. An important part of the analysis is the development of algebraic constraints equivalent to the probabilistic ones. Optimality and duality theory for such models is presented.

In the overview of numerical methods for solving probabilistic optimization problems the emphasis is put on recent numerical methods for nonlinear probabilistically constrained problems based on the optimality and duality theory presented here. The methods provide optimal solutions for convex problems. Otherwise, they solve certain relaxations of the problem and result in suboptimal solutions and upper and lower bounds for the optimal value. Special attention is paid to probabilistic constraints with discrete distributions.

Some numerical approaches via statistical approximations are discussed as well. Numerical techniques of bounding probability in higher dimensional spaces with satisfactory precision are mentioned briefly in the context of discrete distributions. Application of combinatorial techniques in this context is sketched.