Lagrange multipliers, (exact) regularization and error bounds for monotone variational inequalities

Abstract

We examine two central regularization strategies for monotone variational inequalities, the first a direct regularization of the operative monotone mapping, and the second via regularization of the associated dual gap function. A key link in the relationship between the solution sets to these various regularized problems is the idea of exact regularization, which, in turn, is fundamentally associated with the existence of Lagrange multipliers for the regularized variational inequality. A regularization is said to be exact if a solution to the regularized problem is a solution to the unregularized problem for all parameters beyond a certain value. The Lagrange multipliers corresponding to a particular regularization of a variational inequality, on the other hand, are defined via the dual gap function. Our analysis suggests various conceptual, iteratively regularized numerical schemes, for which we provide error bounds, and hence stopping criteria, under the additional assumption that the solution set to the unregularized problem is what we call weakly sharp of order greater than one.

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Acknowledgments

The research of CC and DRL was supported by the Deutsche Forschungsgemeinschaft/German Research Foundation Grant SFB755-A4 and SFB755-C2.

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Correspondence to D. Russell Luke.

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Charitha, C., Dutta, J. & Luke, D.R. Lagrange multipliers, (exact) regularization and error bounds for monotone variational inequalities. Math. Program. 161, 519–549 (2017). https://doi.org/10.1007/s10107-016-1022-6

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Keywords

  • Variational inequality
  • Exact regularization
  • Error bound
  • Gap functional
  • Dual gap functional
  • D-gap function
  • Weak sharp solutions

Mathematics Subject Classification

  • Primary 49J40
  • 47J20
  • Secondary 47H04
  • 49M20
  • 49M37
  • 65K05
  • 90C30