Global Optimization with Polynomials and the Problem of Moments
SIAM Journal on Optimization
SIAM Journal on Optimization
Minimizing Polynomials via Sum of Squares over the Gradient Ideal
Mathematical Programming: Series A and B
Regularization Methods for Semidefinite Programming
SIAM Journal on Optimization
A Newton-CG Augmented Lagrangian Method for Semidefinite Programming
SIAM Journal on Optimization
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We extend the method of Ghasemi and Marshall (SIAM J. Optim. 22(2):460---473, 2012), to obtain a lower bound fgp,M for a multivariate polynomial $f(\mathbf{x}) \in\mathbb{R}[\mathbf {x}]$ of degree ≤2d in n variables x=(x1,驴,xn) on the closed ball $\{ \mathbf{x} \in\mathbb{R}^{n} : \sum x_{i}^{2d} \le M\}$, computable by geometric programming, for any real M. We compare this bound with the (global) lower bound fgp obtained by Ghasemi and Marshall, and also with the hierarchy of lower bounds, computable by semidefinite programming, obtained by Lasserre (SIAM J. Optim. 11(3):796---816, 2001). Our computations show that the bound fgp,M improves on the bound fgp and that the computation of fgp,M, like that of fgp, can be carried out quickly and easily for polynomials having of large number of variables and/or large degree, assuming a reasonable sparsity of coefficients, cases where the corresponding computation using semidefinite programming breaks down.