Joint-range convexity for a pair of inhomogeneous quadratic functions and applications to QP

2015/08/07 by Fabián Flores-Bazán, Felipe Opazo, Flores-Bazán, Fabián +1
Computer Science · Mathematics · #49N10 #49N15 #52A10 #90C20 #90C46 #Advanced Optimization Algorithms Research #FOS: Mathematics #Mathematical Inequalities and Applications #Optimization and Control (math.OC) #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.1508.01612

openalex publication_date 2015/08/07 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We establish various extensions of the convexity Dines theorem for a (joint-range) pair of inhomogeneous quadratic functions. If convexity fails we describe those rays for which the sum of the joint-range and the ray is convex. These results are suitable for dealing nonconvex inhomogeneous quadratic optimization problems under one quadratic equality constraint. As applications of our main results, different sufficient conditions for the validity of S-lemma (a nonstrict version of Finsler's theorem) for inhomogenoeus quadratic functions, is presented. In addition, a new characterization of strong duality under Slater-type condition is established.

Citations

• Non-convex quadratic minimization problems with quadratic constraints: global optimality conditions
• Trust--Region Problems with Linear Inequality Constraints: Exact SDP Relaxation, Global Optimality and Robust Optimization

Related

• Polyak's convexity theorem, Yuan's lemma and S-lemma: extensions and applications
• Optimizing convex functions over nonconvex sets
• Solving a new type of quadratic optimization problem having a joint numerical range constraint
• The Joint Range of Quadratic Mapping on Hilbert Space
• Duality suitable for a class of non-convex optimization problems