A bundle-type auxiliary problem method for solving generalized variational-like inequalities

For generalized variational-like inequalities, by combining the auxiliary principle technique with the bundle idea for nonconvex nonsmooth minimization, we present an implementable iterative method. To make the subproblem easier to solve, even though the preinvex function may not be convex, we still consider using the model similar to the one in [R. Mifflin, A modification and extension of Lemarechal’s algorithm for nonsmooth minimization, Mathematical Programming 17 (1982) 77–90] (which may not be under the preinvex function) to approximate locally the involved preinvex function, and prove that this local approximation is well defined at each iteration of the algorithm, i.e., the construction of this local approximation can terminate in finite steps at each iteration of the proposed algorithm. We not only explain how to construct the approximation, but also prove the weak convergence of the sequence generated by the corresponding algorithm under some conditions. The proposed algorithm is a generalization of the existing algorithm for generalized variational inequalities to generalized variational-like inequalities in some sense, see [T.T. Hue, J.J. Strodiot, V.H. Nguyen, Convergence of the approximate auxiliary problem method for solving generalized variational inequalities, Journal of Optimization Theory and Applications 121 (2004) 119–145].