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Transversality in Variational Analysis

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Abstract

We discuss various aspects of newly developed extension of the classical transversality theory to variational analysis and optimization theory. In particular, we give interpretations in transversality terms of some key results (relating to subdifferential calculus, necessary optimality conditions and linear convergence of alternating projections) and prove a set-valued version of the Thom transversality theorem for semi-algebraic objects.

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Notes

  1. There are of course weaker qualification conditions. The Abadie qualification condition is probably the best known. But all of them, like subtransversality condition in the theorem, guarantee only the existence of a set of Lagrange multipliers with \(\lambda >0\), not normality of the problem.

  2. There is a third equivalent property known as the pseudo-Lipschitz or Aubin property of the inverse mapping \(F^{-1}\). However, we do not need it here.

  3. All subsequent results are valid for a much broader class of definable sets—see, for example, [14].

  4. I am thankful to the referee who brought my attention to a very recent paper [30] that also contains a (different) proof of the implication subtransversality \(\Rightarrow \) intrinsic transversality for convex sets in Euclidean spaces.

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Acknowledgements

I am deeply thankful to the anonymous referee for very attentive reading that allowed to eliminate many misprints and ambiguities in the text.

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  1. Department of Mathematics, Technion, 32000, Haifa, Israel

    Alexander D. Ioffe

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  1. Alexander D. Ioffe
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Correspondence to Alexander D. Ioffe.

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To the memory of Jon Borwein and Vladimir Demyanov.

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Ioffe, A.D. Transversality in Variational Analysis. J Optim Theory Appl 174, 343–366 (2017). https://doi.org/10.1007/s10957-017-1130-3

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