A projected descent method for minimization of weakly semismooth functions over polyhedral sets

Abstract

This study develops a projection-based descent method for minimizing weakly semismooth functions over polyhedral sets, taking an initial step toward extending projection-based descent algorithms to nonsmooth, nonconvex optimization problems. At each iteration, the method considers an inner approximation of the Clarke $\varepsilon$-subdifferential at the current iterate over the feasible region and uses its least-norm element to generate a search direction. An exponential limited backtracking line search is then developed to assess the quality of the generated direction, while a projected variant of Mifflin's line search is proposed to identify nonredundant subgradients that enrich the Clarke $\varepsilon$-subdifferential approximation when necessary. To quantify stationarity, a characterization of stationary points based on the projection operator yields a computable optimality measure for the proposed method. The convergence properties of the proposed method are established under mild assumptions. If the algorithm generates infinitely many serious steps, we identify two subsequences such that every cluster point of each subsequence is stationary. If the number of serious steps is finite, we show that the final serious step generates a stationary point. Numerical experiments demonstrate the efficiency and broad applicability of the proposed method across a wide range of test problems, including applications in image denoising, multiobjective optimization, and data clustering. To extend the applicability of the proposed method to problems with general smooth nonlinear constraints, we propose a heuristic approach based on sequential linearization.