Global Minimization of Normal Quartic Polynomials Based on Global Descent Directions

Abstract

A normal quartic polynomial is a quartic polynomial whose fourth degree term coefficient tensor is positive definite. Its minimization problem is one of the simplest cases of nonconvex global optimization, and has engineering applications. We call a direction a global descent direction of a function at a point if there is another point with a lower function value along this direction. For a normal quartic polynomial, we present a criterion to find a global descent direction at a noncritical point, a saddle point, or a local maximizer. We give sufficient conditions to judge whether a local minimizer is global and give a method for finding a global descent direction at a local, but not global, minimizer. We also give a formula at a critical point and a method at a noncritical point to find a one-dimensional global minimizer along a global descent direction. Based upon these, we propose a global descent algorithm for finding a global minimizer of a normal quartic polynomial when n = 2. For the case $n \ge 3$, we propose an algorithm for finding an $\epsilon$-global minimizer. At each iteration of a second algorithm, a system of constrained nonlinear equations is solved. Numerical tests show that these two algorithms are promising.