Nonconvex homogeneous optimization: a general framework and optimality conditions of first and second-order

Fabián Flores-Bazán ; Adrian Carrillo-Galvez

Minimax theory and its applications, Tome 9 (2024) no. 1

Résumé

This work discusses and analyzes a class of nonconvex homogeneous optimization problems, in which the objective function is a positively homogeneous function with a certain degree, and the constraints set is determined by a single homogeneous function with another degree, and a geometric set which is a (not necessarily convex) closed cone. Once a Lagrangian dual problem is associated, it is provided various characterizations for the validity of strong duality property: one of them is related to the convexity of a certain image of the geometric set involving both homogeneous functions, so revealing a hidden convexity. We also derive a suitable S-lemma. In the case where both functions are of the same degree of homogeneity, a copositive reformulation of the original problem is established. It is also established zero-, first- and second-order optimality conditions; KKT (local or global) optimality, giving rise to the notion of L-eigenvalues with applications to symmetric tensors eigenvalues analysis

Détail

Mots-clés : Nonconvex optimization, homogeneous functions, copositivity, hidden convexity, strong duality, S-lemma