M. Anitescu, "Global Convergence of an Eleastic Mode Approach for a Class of Mathematical Programs with Complementarity Constraints," Preprint ANL/MCS-P1143-0404, April 2004. [pdf]
We prove that any accumulation point of an elastic mode approach, applied to the optimization of a mixed P variational inequality, that approximately solves the relaxed subproblems is a C-stationary point of the problem of optimizing a parametric mixed P variational inequality. If, in addition, the accumulation point satisfies the MPCC-LICQ constant qualification and if the solutions of the subproblem satisfy approximate second-order sufficient conditions, then the limiting point is an M-stationary point. Moreover, if the accumulation point satisfies the upper-level strict complementarity condition, the accumulation point will be a strongly stationary point. If we assume that the penalty function associated with the feasible set of the mathematical program with complementarity constraints has bounded level sets and if the objective function is bounded below, we show that the algorithm will produce bounded iterates and will therefore have at least one accumulation point. We prove that the obstacle problem satisfies our assumptions for both a rigid and a deformable obstacle. The theoretical conclusions are validated by several numerical examples.