On Local Minimizers of Quadratically Constrained Nonconvex Homogeneous Quadratic Optimization with at Most Two Constraints

Mengmeng Song; Hongying Liu; Yong Xia

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arXiv · 2107.05816

On Local Minimizers of Quadratically Constrained Nonconvex Homogeneous Quadratic Optimization with at Most Two Constraints

Mengmeng Song·Hongying Liu·Yong Xia

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Abstract

We study nonconvex homogeneous quadratically constrained quadratic optimization with one or two constraints, denoted by (QQ1) and (QQ2), respectively. (QQ2) contains (QQ1), trust region subproblem (TRS) and ellipsoid regularized total least squares problem as special cases. It is known that there is a necessary and sufficient optimality condition for the global minimizer of (QQ2). In this paper, we first show that any local minimizer of (QQ1) is globally optimal. Unlike its special case (TRS) with at most one local non-global minimizer, (QQ2) may have infinitely many local non-global minimizers. At any local non-global minimizer of (QQ2), both linearly independent constraint qualification and strict complementary condition hold, and the Hessian of the Lagrangian has exactly one negative eigenvalue. As a main contribution, we prove that the standard second-order sufficient optimality condition for any strict local non-global minimizer of (QQ2) remains necessary. Applications and the impossibility of further extension are discussed.

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Mengmeng Song, Hongying Liu, Yong Xia. 2021-07-21. On Local Minimizers of Quadratically Constrained Nonconvex Homogeneous Quadratic Optimization with at Most Two Constraints. https://arxiv.org/abs/2107.05816

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