Chance Constrained Mixed Integer Program: Bilinear and Linear Formulations, and Benders Decomposition

In this paper, we study chance constrained mixed integer program with consideration of recourse decisions and their incurred cost, developed on a finite discrete scenario set. Through studying a non-traditional bilinear mixed integer formulation, we derive its linear counterparts and show that they could be stronger than existing linear formulations. We also develop a variant of Jensen's inequality that extends the one for stochastic program. To solve this challenging problem, we present a variant of Benders decomposition method in bilinear form, which actually provides an easy-to-use algorithm framework for further improvements, along with a few enhancement strategies based on structural properties or Jensen's inequality. Computational study shows that the presented Benders decomposition method, jointly with appropriate enhancement techniques, outperforms a commercial solver by an order of magnitude on solving chance constrained program or detecting its infeasibility

Problem-based optimal scenario generation and reduction in stochastic programming

Scenarios are indispensable ingredients for the numerical solution of stochastic programs. Earlier approaches to optimal scenario generation and reduction are based on stability arguments involving distances of probability measures. In this paper we review those ideas and suggest to make use of stability estimates based only on problem specific data. For linear two-stage stochastic programs we show that the problem-based approach to optimal scenario generation can be reformulated as best approximation problem for the expected recourse function which in turn can be rewritten as a generalized semi-infinite program. We show that the latter is convex if either right-hand sides or costs are random and can be transformed into a semi-infinite program in a number of cases. We also consider problem-based optimal scenario reduction for two-stage models and optimal scenario generation for chance constrained programs. Finally, we discuss problem-based scenario generation for the classical newsvendor problem

Problem-based optimal scenario generation and reduction in stochastic programming

Scenarios are indispensable ingredients for the numerical solution of stochastic programs. Earlier approaches to optimal scenario generation and reduction are based on stability arguments involving distances of probability measures. In this paper we review those ideas and suggest to make use of stability estimates based only on problem specific data. For linear two-stage stochastic programs we show that the problem-based approach to optimal scenario generation can be reformulated as best approximation problem for the expected recourse function which in turn can be rewritten as a generalized semi-infinite program. We show that the latter is convex if either right-hand sides or costs are random and can be transformed into a semi-infinite program in a number of cases. We also consider problem-based optimal scenario reduction for two-stage models and optimal scenario generation for chance constrained programs. Finally, we discuss problem-based scenario generation for the classical newsvendor problem

An augmented lagrangian decomposition method for chance-constrained optimization problems

Joint chance-constrained optimization problems under discrete distributions arise frequently in financial management and business operations. These problems can be reformulated as mixed-integer programs. The size of reformulated integer programs is usually very large even though the original problem is of medium size. This paper studies an augmented Lagrangian decomposition method for finding high-quality feasible solutions of complex optimization problems, including nonconvex chance-constrained problems. Different from the current augmented Lagrangian approaches, the proposed method allows randomness to appear in both the left-hand-side matrix and the right-hand-side vector of the chance constraint. In addition, the proposed method only requires solving a convex subproblem and a 0-1 knapsack subproblem at each iteration. Based on the special structure of the chance constraint, the 0-1 knapsack problem can be computed in quasi-linear time, which keeps the computation for discrete optimization subproblems at a relatively low level. The convergence of the method to a first-order stationary point is established under certain mild conditions. Numerical results are presented in comparison with a set of existing methods in the literature for various real-world models. It is observed that the proposed method compares favorably in terms of the quality of the best feasible solution obtained within a certain time for large-size problems, particularly when the objective function of the problem is nonconvex or the left-hand-side matrix of the constraints is random

On the mixing set with a knapsack constraint

We study a substructure appearing in mixed-integer programming reformulations of chance-constrained programs with stochastic right-hand-sides over a finite discrete distribution, which we call the mixing set with a knapsack constraint. Recently, Luedtke et al. (Math. Program. 122(2):247–272, 2010) and Küçükyavuz (Math Program 132(1):31–56, 2012) studied valid inequalities for such sets. However, most of their results were focused on the equal probabilities case (when the knapsack constraint reduces to a cardinality constraint). In this paper, we focus on the general probabilities case (general knapsack constraint). We characterize the valid inequalities that do not come from the knapsack polytope and use this characterization to generalize the results previously derived for the equal probabilities case. Our results allow for a deep understanding of the relationship that the set under consideration has with the knapsack polytope. Moreover, they allow us to establish benchmarks that can be used to identify when a relaxation will be useful for the considered types of reformulations of chance-constrained programs