Approximate GCDs of polynomials and sparse SOS relaxations
Theoretical Computer Science
Optimization and NP_R-completeness of certain fewnomials
Proceedings of the 2009 conference on Symbolic numeric computation
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Theoretical Computer Science
An iterative scheme for valid polynomial inequality generation in binary polynomial programming
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
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This paper discusses how to find the global minimum of functions that are summations of small polynomials (“small” means involving a small number of variables). Some sparse sum of squares (SOS) techniques are proposed. We compare their computational complexity and lower bounds with prior SOS relaxations. Under certain conditions, we also discuss how to extract the global minimizers from these sparse relaxations. The proposed methods are especially useful in solving sparse polynomial system and nonlinear least squares problems. Numerical experiments are presented which show that the proposed methods significantly improve the computational performance of prior methods for solving these problems. Lastly, we present applications of this sparsity technique in solving polynomial systems derived from nonlinear differential equations and sensor network localization.