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An extension of the Polyak convexity principle with application to nonconvex optimization. (arXiv:1710.03625v1 [math.OC])
来源于:arXiv
The main problem considered in the present paper is to single out classes of
convex sets, whose convexity property is preserved under nonlinear smooth
transformations. Extending an approach due to B.T. Polyak, the present study
focusses on the class of uniformly convex subsets of Banach spaces. As a main
result, a quantitative condition linking the modulus of convexity of such kind
of set, the regularity behaviour around a point of a nonlinear mapping and the
Lipschitz continuity of its derivative is established, which ensures the images
of uniformly convex sets to remain uniformly convex. Applications of the
resulting convexity principle to the existence of solutions, their
characterization and to the Lagrangian duality theory in constrained nonconvex
optimization are then discussed. 查看全文>>