Proximal-Type Algorithms for Solving Nonconvex Mixed Multivalued Quasi-Variational Inequality Problems

We propose iterative algorithms of proximal point type for solving two classes of mixed multivalued quasi-variational inequality problems in real Euclidean spaces involving nonconvex functions, for which no similar algorithms are currently known. Our proposed algorithms combine the proximal type algorithm for solving mixed variational inequalities in a nonconvex framework, the Mann iteration scheme for approximating a fixed point of certain generalized nonexpansive multivalued mappings with the infeasible projection and cutting plane techniques for variational inequalities to generate iterative sequences that converge to a solution of a mixed multivalued quasi-variational inequality problem under mild assumptions. An application to generalized Nash (quasi)equilibrium problems is discussed. Numerical experiments confirm the usability of the introduced algorithms.