A Proximal Approach for Nonsmooth Composite-Constrained Optimization
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We propose a proximal-type algorithm for nonsmooth and nonconvex optimization problems with composite constraints. The constraint is defined by the composition of a locally upper-\(C^2\) outer function with a locally Lipschitz continuous inner mapping. The method is based on an improvement function that balances objective decrease and constraint satisfaction, and on a surrogate model obtained by linearizing the outer function with respect to the composite argument. Each trial point is obtained as a stationary point of a regularized master problem, which may be nonlinear and nonconvex; nevertheless, we assume such a point can be computed efficiently using an off-the-shelf solver. The resulting scheme combines stability-center updates with a sufficient-decrease criterion, while preserving the nonlinear structure of the inner mapping. We prove that every accumulation point of the serious iterates is critical for the original problem and, under a suitable constraint qualification, satisfies a generalized KKT condition. Numerical experiments on chance-constrained optimization problems demonstrate the practical performance and robustness of the proposed method.