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 ≥ 3, we propose an algorithm for finding an ε-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.
Original language | English |
---|---|
Pages (from-to) | 275-302 |
Number of pages | 28 |
Journal | SIAM Journal on Optimization |
Volume | 15 |
Issue number | 1 |
DOIs | |
Publication status | Published - 29 Mar 2005 |
Keywords
- Global optimization
- Normal quartic polynomial
- Tensor
ASJC Scopus subject areas
- Theoretical Computer Science
- Software