(426e) Solving Joint Chance Constrained Problem Via Robust Optimization Approximation

Chance/probabilistic constraint enforces probabilistic guarantee on constraint satisfaction and provides an important way of quantifying solution reliability for optimization under parameter uncertainty. Chance constrained optimization problem is generally very challenging due to the difficulties in checking the feasibility of the probabilistic constraint and the nonconvexity of the feasible region. As a result, chance constrained problems are normally solved through approximation except for the cases that a deterministic equivalent model can be obtained (e.g., normal distribution) [1][2].

As an important methodology for dealing with optimization problems with data uncertainty, uncertainty set induced robust optimization has been successfully used to safely approximate individual chance constraint in the past [3][4][5]. In set induced robust optimization, the uncertain data is assumed to be varying in an adjustable uncertainty set, and the aim is to choose the best solution among those “immunized” against data uncertainty.

While individual chance constraint focus on probabilistic guarantee on individual constraint satisfaction, joint chance constraint enforces that several constraints are satisfied simultaneously under parameter uncertainty. Thus, it is useful in modeling uncertainty correlation in different constraints. However, it becomes more complicated to handle joint chance constrained optimization problem, and its solution is not as satisfactory as individual chance constraint so far [6]. In this work, we extend the robust optimization approximation framework to the solution of joint chance constrained problem. Specifically, joint chance constraint is converted to single chance constraint problem through the introduction of inner maximization problem, and then safely approximated with the aid of probability inequalities. Different robust optimization formulations are then derived for joint chance constraint. To illustrate the proposed robust optimization approximation framework, numerical examples are studied and the solution is compared to the traditional decomposition method for approximating joint chance constraint.

References:

[1] A. Nemirovski, A. Shapiro. Convex approximations of chance constrained programs. SIAM J. Optim. 2006, 17(4), 969

[2] S. Ahmed and A. Shapiro. "Solving chance-constrained stochastic programs via sampling and integer programming," in Tutorials in Operations Research, Z.-L. Chen and S. Raghavan (eds.), INFORMS, 2008

[3] Z. Li, R. Ding, C.A. Floudas. A comparative theoretical and computational study on robust counterpart optimization: I. Robust linear optimization and robust mixed integer linear optimization. Industrial & Engineering Chemistry Research, 2011, 50, 10567-10603.

[4] Z. Li, Q. Tang, C.A. Floudas. A comparative theoretical and computational study on robust counterpart optimization: II. Probabilistic guarantees on constraint satisfaction. Industrial & Engineering Chemistry Research. 2012, 51, 6769-6788.

[5] Z. Li, C.A. Floudas. A Comparative Theoretical and Computational Study on Robust Counterpart Optimization: III. Improving the Quality of Robust Solutions. Industrial & Engineering Chemistry Research. Under review. 2014

[6] Chen, W., Sim, M., Sun, J., & Teo, C. P. From CVaR to uncertainty set: Implications in joint chance-constrained optimization. Operations research, 2010, 58, 470-485.