Theoretical and Numerical Comparison of Relaxation Methods for Mathematical Programs with Complementarity Constraints

Theoretical and Numerical Comparison of Relaxation Methods for Mathematical Programs with Complementarity Constraints. Hoheisel, T., Kanzow, C., & Schwartz, A.
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Mathematical programs with equilibrium constraints (MPECs) are difﬁcult optimization problems whose feasible sets do not satisfy most of the standard constraint qualiﬁcations. Hence MPECs cause difﬁculties both from a theoretical and a numerical point of view. As a consequence, a number of MPEC-tailored solution methods have been suggested during the last decade which are known to converge under suitable assumptions. Among these MPEC-tailored solution schemes, the relaxation methods are certainly one of the most prominent class of solution methods. Several different relaxation schemes are available in the meantime, and the aim of this paper is to provide a theoretical and numerical comparison of these schemes. More precisely, in the theoretical part, we improve the convergence theorems of several existing relaxation methods. There, we also take a closer look at the properties of the feasible sets of the relaxed problems and show which standard constraint qualiﬁcations are satisﬁed for these relaxed problems. Finally, the numerical comparison is based on the MacMPEC test problem collection.

@article{hoheisel_theoretical_nodate,
	title = {Theoretical and {Numerical} {Comparison} of {Relaxation} {Methods} for {Mathematical} {Programs} with {Complementarity} {Constraints}},
	doi = {10.1007/s10107-011-0488-5},
	abstract = {Mathematical programs with equilibrium constraints (MPECs) are difﬁcult optimization problems whose feasible sets do not satisfy most of the standard constraint qualiﬁcations. Hence MPECs cause difﬁculties both from a theoretical and a numerical point of view. As a consequence, a number of MPEC-tailored solution methods have been suggested during the last decade which are known to converge under suitable assumptions. Among these MPEC-tailored solution schemes, the relaxation methods are certainly one of the most prominent class of solution methods. Several different relaxation schemes are available in the meantime, and the aim of this paper is to provide a theoretical and numerical comparison of these schemes. More precisely, in the theoretical part, we improve the convergence theorems of several existing relaxation methods. There, we also take a closer look at the properties of the feasible sets of the relaxed problems and show which standard constraint qualiﬁcations are satisﬁed for these relaxed problems. Finally, the numerical comparison is based on the MacMPEC test problem collection.},
	language = {en},
	author = {Hoheisel, Tim and Kanzow, Christian and Schwartz, Alexandra},
	pages = {32},
}

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