Distributionally Robust Joint Chance Constraints Optimization

Chance constraints is a classical modeling method for the problems with random variables.However, it is difficult to solve chance constraints due to the multiple integral involved withcomputing the probability and the non-convexity. A method for deal with chance constraintsis to find some computationally tractable convex constraints whose optimal solution can satisfythe chance constraints and objective value is close to the optimal. This convex constraints arecalled safe approximation of the chance constraints. This paper consider the distributionallyrobust chance constraints optimization. Assume that the first and second order moments of therandom variables are known. For individual chance constraints, we review a safe approximationby Worst-Case Conditional Value-at-Risk (WC-CVaR). For joint chance constraints, we reviewtwo safe approximations, one is given by Bonferroni inequality, another is given by WC-CVaR.All of these approximations is semidefinite constraints.Then we present a new approximation in terms of semidefinite constraints for joint chanceconstraints. Furthermore, we develop our approximation to the case of injecting the supportinformation of the random variables. Similar to the WC-CVaR approximation, our approxima-tion is deduced from the moment problem corresponding to the chance constraints, convert it tocertain tractable convex constraints by the dual technology. we prove that our approximation isin fact equivalent to the chance constraints as well as the WC-CVaR approximation when theconstraint functions are linear in the random variables. Thus these two approximation shouldget the same optimal objective value in theory, it has proved by the subsequent numerical tests.In the numerical experimentation, we solve the dynamic water reservoir control problem by thethree approximation for joint chance constraints. The result show that the WC-CVaR approx-imation and our approximation has almost same optimal objective value, and far surpass theBonferroni’s. The running speed of our approximation is much slower than the Bonferroni’s,but is superior to the WC-CVaR approximation.