An extension of the Polyak convexity principle with application to nonconvex optimization

zbMATH Open1458.52002arXiv1710.03625MaRDI QIDQ5151658

Publication date: 22 February 2021

Abstract: 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.

Full work available at URL: https://arxiv.org/abs/1710.03625

zbMATH Keywords

constrained optimization modulus of convexity strong duality uniformly convex sets openness at a linear rate/metric regularity

Mathematics Subject Classification ID

Nonconvex programming, global optimization (90C26) Optimality conditions and duality in mathematical programming (90C46) Programming in abstract spaces (90C48) Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry) (52A21) Convex sets without dimension restrictions (aspects of convex geometry) (52A05)