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Standard BiQuadratic Optimization Problems and Unconstrained Polynomial Reformulations
, 2009
"... Abstract. A socalled Standard BiQuadratic Optimization Problem (StBQP) consists in minimizing a biquadratic form over the Cartesian product of two simplices (so this is different from a BiStandard QP where a quadratic function is minimized over the same set). An application example arises in por ..."
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Abstract. A socalled Standard BiQuadratic Optimization Problem (StBQP) consists in minimizing a biquadratic form over the Cartesian product of two simplices (so this is different from a BiStandard QP where a quadratic function is minimized over the same set). An application example arises in portfolio selection. In this paper we present a biquartic formulation of StBQP, in order to get rid of the sign constraints. We study the first and secondorder optimality conditions of the original StBQP and the reformulated biquartic problem over the product of two Euclidean spheres. Furthermore, we discuss the onetoone 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. Key Words. Polynomial optimization, standard simplex, biquartic optimization, optimality conditions, penalty function.
and unconstrained polynomial reformulations
, 2009
"... Standard biquadratic optimization problems ..."
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