Standard bi‐quadratic optimization problems and unconstrained polynomial reformulations

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.