Convexity Conditions and the Legendre-Fenchel Transform for the Product of Finitely Many Positive Definite Quadratic Forms

Yun-Bin Zhao

Research output: Contribution to journalArticle

2 Citations (Scopus)
364 Downloads (Pure)

Abstract

While the product of finitely many convex functions has been investigated in the field of global optimization, some fundamental issues such as the convexity condition and the Legendre-Fenchel transform for the product function remain unresolved. Focusing on quadratic forms, this paper is aimed at addressing the question: When is the product of finitely many positive definite quadratic forms convex, and what is the Legendre-Fenchel transform for it? First, we show that the convexity of the product is determined intrinsically by the condition number of so-called 'scaled matrices' associated with quadratic forms involved. The main result claims that if the condition number of these scaled matrices are bounded above by an explicit constant (which depends only on the number of quadratic forms involved), then the product function is convex. Second, we prove that the Legendre-Fenchel transform for the product of positive definite quadratic forms can be expressed, and the computation of the transform amounts to finding the solution to a system of equations (or equally, finding a Brouwer's fixed point of a mapping) with a special structure. Thus, a broader question than the open "Question 11" in Hiriart-Urruty (SIAM Rev. 49, 225-273, 2007) is addressed in this paper.
Original languageEnglish
Pages (from-to)411-434
Number of pages24
JournalApplied Mathematics & Optimization
Volume62
Issue number3
DOIs
Publication statusPublished - 1 Dec 2010

Keywords

  • Condition numbers
  • Positive definite matrices
  • Convex analysis
  • Matrix analysis
  • Legendre-Fenchel transform
  • Quadratic forms

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