Nonsmooth Techniques for Computing Projected Solutions of Quasiequilibria via Gap Functions

Projected solutions to a quasiequilibrium problem allow overcoming the possible lack of solutions when the constraining set-valued map is not a self-map. This paper aims at providing a descent algorithm for computing projected solutions by relying on a reformulation of the problem as a nonsmooth optimization problem. The nonsmoothness of the gap function can be dealt with successfully through the nonexpansiveness of the projection and tools such as Clarke subdifferentials. Nonetheless, some additional difficulties arise since the projection brings in nonsmoothness also in constraints that are provided by differentiable bifunctions. Monotonicity assumptions on the constraints have to cope with this further issue both to devise the algorithm and prove its convergence. Preliminary numerical tests show a promising behaviour of the algorithm.