An algorithm is presented for solving nonlinear optimization problems with chance constraints, i.e., those in which a constraint involving an uncertain parameter must be satisfied with at least a minimum probability. In particular, the algorithm is designed to solve cardinality-constrained nonlinear optimization problems that arise in sample average approximations of chance-constrained problems, as well as in other applications in which it is only desired to enforce a minimum number of constraints. The algorithm employs a novel penalty function, which is minimized sequentially by solving quadratic optimization subproblems with linear cardinality constraints. Properties of minimizers of the penalty function in relation to minimizers of the corresponding nonlinear optimization problem are presented, and convergence of the proposed algorithm to a stationary point of the penalty function is proved. The effectiveness of the algorithm is demonstrated through numerical experiments with a nonlinear cash flow problem.
F.E.Curtis, A.Waechter,and V.M.Zavala. A Sequential Algorithm for Solving Nonlinear Optimization Problems with Chance Constraints. SIAM Journal on Optimization, 28(1):930–958, 2018.