The price of inexactness: convergence properties of relaxation methods for mathematical programs with complementarity constraints revisited.

*(English)*Zbl 1344.90058Summary: 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 S. Scholtes [SIAM J. Optim. 11, No. 4, 918–936 (2001; Zbl 1010.90086)], 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 [loc. cit.] 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.

##### MSC:

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |

65K05 | Numerical mathematical programming methods |

49M37 | Numerical methods based on nonlinear programming |

##### Keywords:

mathematical programs with complementarity constraints; mathematical programs with equilibrium constraints; global convergence; KKT-points; stationary points; strong stationarity; M-stationarity; C-stationarity; weak stationarity; inexact relaxation methods; inexact regularization methods
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\textit{C. Kanzow} and \textit{A. Schwartz}, Math. Oper. Res. 40, No. 2, 253--275 (2015; Zbl 1344.90058)

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