An Algorithm for Constrained Optimization with Semismooth Functions

We present an implementable algorithm for solving constrained optimization problems defined by functions that are not everywhere differentiable. The method is based on combining, modifying and extending the nonsmooth optimization work of Wolfe, Lemarechal, Feuer, Poljak, and Merrill. It can be thought of as a generalized reset conjugate gradient algorithm. We also introduce the class of weakly upper semismooth functions. These functions are locally Lipschitz and have a semicontinuous relationship between their generalized gradient sets and their directional derivatives. The algorithm is shown to converge to stationary points of the optimization problem if the objective and constraint functions are weakly upper semismooth. Such points are optimal points if the problem functions are also semiconvex and a constraint qualification is satisfied. Under stronger convexity assumptions, bounds on the deviation from optimality of the algorithm iterates are given