The Price of Inexactness: Convergence Properties of Relaxation Methods for Mathematical Programs with Complementarity Constraints Revisited

Mathematical programs with equilibrium (or complementarity) constraints, MPECs for short, form a difficult class of optimization problems. The feasible set has a very special structure and violates most of the standard constraint qualifications. Therefore, one typically applies specialized algorithms in order to solve MPECs. One prominent class of specialized algorithms is the relaxation (or regularization) methods. The first relaxation method for MPECs is due to Scholtes [35] [Scholtes S (2001) Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM Journal on Optimization 11:918–936.], but in the meantime, there exists a number of different regularization schemes that try to relax the difficult constraints in different ways. Some of these more recent schemes have better theoretical properties than does the original method by Scholtes. Nevertheless, numerical experience shows that the Scholtes relaxation method is still among the fastest and most reliable ones. To give a possible explanation for this, we consider that, numerically, the regularized subproblems are not solved exactly. In this light, we analyze the convergence properties of a number of relaxation schemes and study the impact of inexactly solved subproblems on the kind of stationarity we can expect in a limit point. Surprisingly, it turns out that the inexact version of Scholtes' method has the same convergence properties as its exact counterpart, whereas most of the other relaxation schemes lose a lot of their original properties.