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Journal of Convex Analysis 17 (2010), No. 2, 673--680
Copyright Heldermann Verlag 2010

Necessary Conditions for Local Optimality in Difference-of-Convex Programming

Immanuel M. Bomze
Dept. of Statistics, University of Vienna, Brünner Str. 72, 1210 Vienna, Austria

Claude Lemaréchal
INRIA Grenoble-Rhône-Alpes, 655 Avenue de l'Europe,, 38334 Saint Ismier - Montbonnot, France

Using ε-subdifferential calculus for difference-of-convex (d.c.) optimization, Dür proposed a condition sufficient for local optimality, and showed that this condition is not necessary in general. Here it is proved that whenever the convex part is strongly convex, this condition is also necessary. Strong convexity can always be ensured by changing the given d.c. decomposition slightly. This approach also allows for a formulation with perturbed ε-subdifferentials which involves only the original d.c. decomposition, even without imposing strong convexity. We relate this result with another inclusion condition on perturbed ε-subdifferentials, which even can serve as a quantitative version of a criterion both necessary and sufficient for local optimality.

Keywords: Approximate subdifferential, non-smooth optimization, optimality condition, strong convexity.

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