Standard bi-quadratic optimization problems and unconstrained polynomial reformulations

  • Authors:

  • Immanuel M. Bomze;Chen Ling;Liqun Qi;Xinzhen Zhang

  • Affiliations:

  • Department of Statistics and Operations Research, University of Vienna, Vienna, Austria;School of Science, Hangzhou Dianzi University, Hangzhou, China 310018;Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong;Department of Mathematics, School of Science, Tianjin University, Tianjin, China 300072

  • Venue:
  • Journal of Global Optimization
  • Year:
  • 2012

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Abstract

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.