Minimax Theory and its Applications 09 (2024), No. 1, 085--115
Copyright Heldermann Verlag 2024
Nonconvex Homogeneous Optimization: a General Framework and Optimality Conditions of First and Second-Order
Fabián Flores-Bazán
Dep. de Ingeniería Matemática, Universidad de Concepción, Concepción, Chile
fflores@ing-mat.udec.cl
Adrian Carrillo-Galvez
Dep. de Ingeniería Matemática, Universidad de Concepción, Concepción, Chile
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.
Keywords: Nonconvex optimization, homogeneous functions, copositivity, hidden convexity, strong duality, S-lemma.
MSC: 90C20, 90C26, 90C34, 90C46, 49N15.
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