A Sequential Algorithm for Solving Nonlinear Optimization Problems with Chance Constraints

Abstract

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.

A scalable stochastic programming approach for the design of flexible systems

“…Specifically, different combinations of active constraints give the same or a very similar SF index. This behavior has been recently reported in [3], where the authors note that chance constraints give rise to a wide range of local minima with similar values. This behavior becomes more evident when one moves the SF index into the objective (as opposed to imposing it in the constraints).…”

Chance constrained optimization of elliptic PDE systems with a smoothing convex approximation

“…This work extends our recent research findings in [26,27] with respect to finite dimensional chance constrained optimization problem to infinite dimensional CCPDE problems. We point out to our readers that recent numerical considerations in [21,31] w.r.t. finite-dimensional (CCOPT) are also potential extendable for CCPDE.…”

Machine learning approach to chance-constrained problems: An algorithm based on the stochastic gradient descent

“…Since the generation of all cuts is difficult, the authors proposed a practical heuristic approach. [9] introduced a sequential algorithm for solving nonlinear chance constrained problems. The method is based on an exact penalty function which is minimized sequentially by solving quadratic optimization subproblems with linear cardinality constraints.…”