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Standard Bi-Quadratic Optimization Problems and Unconstrained Polynomial Reformulations

Immanuel M. Bomze (immanuel.bomze***at***univie.ac.at)
Chen Ling (linghz***at***163.com)
Liqun Qi (maqilq***at***polyu.edu.hk)
Xinzhen Zhang (xzzhang***at***yahoo.cn)

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.

Keywords: Polynomial optimization, standard simplex, bi-quartic optimization, optimality conditions, penalty function

Category 1: Global Optimization (Theory )

Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization )

Citation: Technical Report TR-ISDS 2009-03, Department of Statistics and Decision Support Systems, University of Vienna, Austria (August 2009)

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Entry Submitted: 08/08/2009
Entry Accepted: 08/10/2009
Entry Last Modified: 08/12/2009

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