Abstract
The results on regularity behavior of solutions to variational inequalities over polyhedral sets proved in a series of papers by Robinson, Ralph and Dontchev-Rockafellar in the 90s has long become classics of variational analysis. But the available proofs are very complicated and practically do not use techniques of variational analysis. The only exception is the proof by Dontchev and Rockafellar of their “critical face” regularity criterion. In the paper we offer a different approach completely based on polyhedral geometry and a few basic principles of metric regularity theory. It leads to new proofs, that look simpler and shorter, and in addition gives some clarifying geometrical information.
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Notes
Facchinei and Pang write in the beginning of p. 372 to explain the absence of a general proof of the implication “coherent orientation \(\Rightarrow \) single-valuedness” after proving it for \(n=2\): “the general case \(n\ge 3\) is much more technical and difficult ...”. Likewise, Dontchev and Rockafellar write after the statement of Theorem 4H.1 that “the most important part of this theorem that, for the mapping \(f +N(C,\cdot )\) with a smooth f and a polyhedral convex set C, metric regularity is equivalent to strong metric regularity, will not be proved here in full generality. To prove this fact we need tools that go beyond the scope of this book”.
Note also that there are variational inequalities with non-polyhedral C for which regularity and strong regularity are equivalent, for instance when C is an arbitrary convex set and A is symmetric positive semi-definite. This follows from the fact that for a maximal monotone operator metric regularity implies strong regularity—see [2]. Another relevant example is the critical point mapping in cone constrained convex problems [8], Corollary 1.
I am thankful to an anonimous reviewer who has informed me about this paper of which I was unaware.
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I wish to thank the reviewers for thoughtful comments.
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This research was supported in part by the Israel-USA BSF under the grant 2014241-2.
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Ioffe, A.D. On variational inequalities over polyhedral sets. Math. Program. 168, 261–278 (2018). https://doi.org/10.1007/s10107-016-1077-4
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DOI: https://doi.org/10.1007/s10107-016-1077-4