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On the Subdifferentiability of the Difference of Two Functions and Local Minimization

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Abstract

We set up a formula for the Fréchet and ε-Fréchet subdifferentials of the difference of two convex functions. We even extend it to the difference of two approximately starshaped functions. As a consequence of this formula, we give necessary and sufficient conditions for local optimality in nonconvex optimization. Our analysis relies on the notion of gap continuity of multivalued maps and involves concepts of independent interest such as the notions of blunt and sharp minimizers and the notion of equi-subdifferentiability.

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Authors and Affiliations

  1. Faculté des Sciences et Techniques Guéliz, B.P. 549, Marrakech, Morocco

    Tijani Amahroq & Aïcha Syam

  2. Faculté des Sciences, Mathématiques UMR CNRS 5142, B.P. 1155, 64013, PAU Cedex, France

    Jean-Paul Penot

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  1. Tijani Amahroq
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  2. Jean-Paul Penot
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Amahroq, T., Penot, JP. & Syam, A. On the Subdifferentiability of the Difference of Two Functions and Local Minimization. Set-Valued Anal 16, 413–427 (2008). https://doi.org/10.1007/s11228-008-0085-9

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  • DOI: https://doi.org/10.1007/s11228-008-0085-9

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