A polynomial optimization framework for polynomial quasi-variational inequalities with Moment-SOS relaxations

We consider quasi-variational inequality problems (QVI) given by polynomial functions. By applying Lagrange multiplier expressions, we formulate polynomial optimization problems whose minimizers are KKT points for the QVI. Then, feasible extensions are exploited to preclude KKT points that are not solutions. Moment-SOS relaxations are incorporated to solve the polynomial optimization problems in our methods. Under certain conditions, our approach guarantees to find a solution to the QVI or detect the nonexistence of solutions.