A Lasry-Lions envelope approach for mathematical programs with complementarity constraints

We propose a homotopy method for solving mathematical programs with complementarity constraints (CCs). The indicator function of the CCs is relaxed by the Lasry--Lions double envelope, an extension of the Moreau envelope that enjoys an additional smoothness property, making it amenable to fast optimization algorithms. The proposed algorithm mimics the behavior of homotopy methods for systems of nonlinear equations or penalty methods for constrained optimization: it solves a sequence of smooth subproblems that progressively approximate the original problem, using the solution of each subproblem as the starting point for the next one. In the limiting setting, we establish the convergence to Mordukhovich and Clarke stationary points. We also provide a worst-case complexity analysis for computing an approximate stationary point. Preliminary numerical results on a suite of benchmark problems demonstrate the effectiveness of the proposed approach.