On deterministic reformulations of distributionally robust joint chance constrained optimization problems

A joint chance constrained optimization problem involves multiple uncertain constraints, i.e., constraints with stochastic parameters, that are jointly required to be satisfied with probability exceeding a prespecified threshold. In a distributionally robust joint chance constrained optimization problem (DRCCP), the joint chance constraint is required to hold for all probability distributions of the stochastic parameters from a given ambiguity set. In this work, we consider DRCCP involving convex nonlinear uncertain constraints and an ambiguity set specified by convex moment constraints. We investigate deterministic reformulations of such problems and conditions under which such deterministic reformulations are convex. In particular we show that a DRCCP can be reformulated as a convex program if one the following conditions hold: (i) there is a single uncertain constraint, (ii) the ambiguity set is defined by a single moment constraint, (iii) the ambiguity set is defined by linear moment constraints, and (iv) the uncertain and moment constraints are positively homogeneous with respect to uncertain parameters. We further show that if the decision variables are binary and the uncertain constraints are linear then a DRCCP can be reformulated as a deterministic mixed integer convex program. Finally, we present a numerical study to illustrate that the proposed mixed integer convex reformulation can be solved efficiently by existing solvers. ∗[email protected] †[email protected]