Standard bi-quadratic optimization problems and unconstrained polynomial reformulations

Immanuel M. Bomze, Chen Ling, Liqun Qi, Xinzhen Zhang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

8 Citations (Scopus)


A so-called Standard Bi-Quadratic Optimization Problem (StBQP) consists in minimizing a bi-quadratic form over the Cartesian product of two simplices (so this is different from a Bi-Standard QP where a quadratic function is minimized over the same set). An application example arises in portfolio selection. In this paper we present a bi-quartic formulation of StBQP, in order to get rid of the sign constraints. We study the first- and second-order optimality conditions of the original StBQP and the reformulated bi-quartic problem over the product of two Euclidean spheres. Furthermore, we discuss the one-to-one correspondence between the global/local solutions of StBQP and the global/local solutions of the reformulation. We introduce a continuously differentiable penalty function. Based upon this, the original problem is converted into the problem of locating an unconstrained global minimizer of a (specially structured) polynomial of degree eight.
Original languageEnglish
Pages (from-to)663-687
Number of pages25
JournalJournal of Global Optimization
Issue number4
Publication statusPublished - 1 Apr 2012


  • Bi-quartic optimization
  • Optimality conditions
  • Penalty function
  • Polynomial optimization
  • Standard simplex

ASJC Scopus subject areas

  • Computer Science Applications
  • Control and Optimization
  • Management Science and Operations Research
  • Applied Mathematics

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