Extended SQP Methods in Nonsmooth Difference Programming Applied to Problems with Variational Inequality Constraints

This paper explores a new class of constrained difference programming problems, where the objective and constraints are formulated as differences of functions without requiring their convexity. To investigate such problems, novel variants of the extended sequential quadratic method are introduced. These algorithms iteratively solve strongly convex quadratic subproblems constructed via linear approximations of the given data by using their gradients and subgradients. The convergence of the proposed methods is rigorously analyzed by employing, in particular, the Polyak–Łojasiewicz–Kurdyka property that ensures global convergence for various classes of functions in the problem formulation, for example, semialgebraic ones. The original framework is further extended to address difference programming problems with variational inequality (VI) constraints. By reformulating VI constraints via regularized gap functions, such problems are naturally embedded into constrained difference programming that leads us to direct applications of the proposed algorithms. Numerical experiments for the class of continuous network design problems demonstrate the efficiency of the new methods. Funding: B. S. Mordukhovich is partially supported by the U.S. National Science Foundation [Grant DMS-2204519] and the Australian Research Council [Discovery Project DP250101112]. S. Zeng is supported by the National Natural Science Foundation of China [Grant 12501429] and the Shenzhen Fundamental Research Program [Grant 20250530150024003]. The research of J. Zhang was supported by the National Key R&D Program of China [Grant 2023YFA1011400] and the National Natural Science Foundation of China [Grant 12326605].