English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We study nonconvex homogeneous quadratically constrained quadraticoptimization with one or two constraints, denoted by (QQ1) and (QQ2),respectively. (QQ2) contains (QQ1), trust region subproblem (TRS) and ellipsoidregularized total least squares problem as special cases. It is known thatthere is a necessary and sufficient optimality condition for the globalminimizer 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 localnon-global minimizer, (QQ2) may have infinitely many local non-globalminimizers. At any local non-global minimizer of (QQ2), both linearlyindependent constraint qualification and strict complementary condition hold,and the Hessian of the Lagrangian has exactly one negative eigenvalue. As amain contribution, we prove that the standard second-order sufficientoptimality condition for any strict local non-global minimizer of (QQ2) remainsnecessary. Applications and the impossibility of further extension arediscussed.

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