On Using the Elastic Mode in Nonlinear Programming Approaches to Mathematical Programs with Complementarity Constraints

We investigate the possibility of solving mathematical programs with complementarity constraints (MPCCs) using algorithms and procedures of smooth nonlinear programming. Although MPCCs do not satisfy a constraint qualification, we establish sufficient conditions for their Lagrange multiplier set to be nonempty. MPCCs that have nonempty Lagrange multiplier sets and that satisfy the quadratic growth condition can be approached by the elastic mode with a bounded penalty parameter. In this context, the elastic mode transforms MPCC into a nonlinear program with additional variables that has an isolated stationary point and local minimum at the solution of the original problem, which in turn makes it approachable by sequential quadratic programming algorithms. One such algorithm is shown to achieve local linear convergence once the problem is relaxed. Under stronger conditions, we also prove superlinear convergence to the solution of an MPCC using an adaptive elastic mode approach for a sequential quadratic programming algorithm recently analyzed in an MPCC context by Fletcher and al. [16]. Our assumptions are more general since we do not use a critical assumption from that reference. In addition, we show that the elastic parameter update rule will not interfere locally with the super linear convergence once the penalty parameter is appropriately chosen.