The Price of Inexactness: Convergence Properties of Relaxation Methods for Mathematical Programs with Equilibrium 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 are the relaxation (or regularization) methods. The first relaxation method for MPECs is due to Scholtes [SIAM Journal on Optimization 11, 2001, pp. 918–936 ], but in the meantime, there exist a number of different regularization schemes which try to relax the difficult constraints in different ways. Among the most recent examples for such methods are the ones from Kadrani, Dussault, and Benchakroun [SIAM Journal on Optimization 20, 2009, pp. 78–103 ] and Kanzow and Schwartz [SIAM Journal on Optimization, to appear ]. Surprisingly, although these recent methods have better theoretical properties than Scholtes’ relaxation, numerical comparisons show that this method is still among the fastest and most reliable ones, see for example Hoheisel et al. [Mathematical Programming 137, 2013, pp. 257–288 ]. To give a possible explanation for this, we consider the fact 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.