ON THE CONVEXITY FOR THE RANGE SET OF TWO QUADRATIC FUNCTIONS

Huu Quang Nguyen, Ya Chi Chu, Ruey Lin Sheu

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

Abstract

Given n × n symmetric matrices A and B, Dines in 1941 proved that the joint range set {(xTAx, xTBx) | x ∈ ℝn} is always convex. Our paper is concerned with non-homogeneous extension of the Dines theorem for the range set R(f, g) = {(f(x), g(x)) | x ∈ ℝn}, f(x) = xTAx + 2aTx + a0 and g(x) = xTBx + 2bTx + b0. We show that R(f, g) is convex if, and only if, any pair of level sets, {x ∈ ℝn|f (x) = α} and {x ∈ ℝn|g(x) = β}, do not separate each other. With the novel geometric concept about separation, we provide a polynomial-time procedure to practically check whether a given R(f, g) is convex or not.

Original languageEnglish
Pages (from-to)575-592
Number of pages18
JournalJournal of Industrial and Management Optimization
Volume18
Issue number1
DOIs
Publication statusPublished - 2022 Jan

All Science Journal Classification (ASJC) codes

  • Business and International Management
  • Strategy and Management
  • Control and Optimization
  • Applied Mathematics

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