An implementable proximal-type method for computing critical points to minimization problems with a nonsmooth and nonconvex constraint
Résumé
Modern optimization problems from stochastic optimization, machine learning, uncertainty quantification, and others often involve functions that fail to be smooth and convex. Nevertheless, many such problems possess some structure that can and should be exploited in numerical optimization. This work investigates implementable necessary optimality conditions for a broad class of nonsmooth and nonconvex optimization problems involving objective and constraint functions that can be expressed as a finite sum of pointwise maxima of finitely many weakly-concave functions. Such a setting is general enough to accommodate nonconvex reliability-based optimization problems for which mathematically sound algorithms are scarce. Application areas include energy management, electric power systems, transportation, engineering design, and manufacturing. Our main contribution is a proximal-type algorithm with convergence guarantees to critical points. Our approach does not require penalization nor a feasible initial point. Numerical experiments on some stochastic reliability-based optimization problems illustrate the practical performance of the method.