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The first-order optimality conditions of convexly constrainednonconvex-nonconcave min-max optimization problems formulate variationalinequality problems, which are equivalent to a system of nonsmooth equations.In this paper, we propose a quasi-Newton subspace trust region (QNSTR)algorithm for the least-square problem defined by the smoothing approximationof the nonsmooth equation. Based on the structure of the least-square problem,we use an adaptive quasi-Newton formula to approximate the Hessian matrix andsolve a low-dimensional strongly convex quadratic program with ellipseconstraints in a subspace at each step of QNSTR algorithm. According to thestructure of the adaptive quasi-Newton formula and the subspace technique, thestrongly convex quadratic program at each step can be solved efficiently. Weprove the global convergence of QNSTR algorithm to an $\epsilon$-first-orderstationary point of the min-max optimization problem. Moreover, we presentnumerical results of QNSTR algorithm with different subspaces for the mixedgenerative adversarial networks in eye image segmentation using real data toshow the efficiency and effectiveness of QNSTR algorithm for solving largescale min-max optimization problems.

A Quasi-Newton Subspace Trust Region Algorithm for Least-square Problems in Min-max Optimization