Distinct solutions of finite-dimensional complementarity problems

Complementarity problems often permit distinct solutions, a fact of major significance in optimization, game theory and other fields. In this paper, we develop a numerical technique for computing multiple isolated solutions of complementarity problems, starting from the same initial guess. This technique, called deflation, is applied in conjunction with existing algorithms that reformulate the complementarity problem as the rootfinding problem of a semismooth residual. After one solution is found, the idea of deflation is to apply operators to the arguments of the corresponding semismooth reformulation to ensure that solvers will not converge to that same solution again. This ensures that if the semismooth solver is restarted from the same initial guess and it converges, it will converge to a different solution. We prove theoretical results on the effectiveness of the method, and apply it to several difficult finite-dimensional complementarity problems from the literature. While deflation is not guaranteed to find all solutions, for every problem considered with a finite number of solutions, we identify initial guesses from which all known solutions are computed with deflation.