Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/16215
Title: Standard bi-quadratic optimization problems and unconstrained polynomial reformulations
Authors: Bomze, IM
Ling, C
Qi, L 
Zhang, X
Issue Date: 2012
Source: Journal of global optimization, 2012, v. 52, no. 4, p. 663-687
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: Bi-quartic optimization
Optimality conditions
Penalty function
Polynomial optimization
Standard simplex
Publisher: Springer
Journal: Journal of global optimization 
ISSN: 0925-5001
EISSN: 1573-2916
DOI: 10.1007/s10898-011-9710-5
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