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Approximation algorithms for homogeneous polynomial optimization with quadratic constraints

MATHEMATICAL PROGRAMMING 2010年第2期125卷 353-383页

作者： He, Simai Li, Zhening Zhang, Shuzhong Chinese Univ Hong Kong Dept Syst Engn & Engn Management Shatin Hong Kong Peoples R China City Univ Hong Kong Dept Management Sci Kowloon Tong Hong Kong Peoples R China

In this paper, we consider approximation algorithms for optimizing a generic multi-variate homogeneous polynomial function, subject to homogeneous quadratic constraints. Such optimization models have wide applications, e.g., in signal processing, magnetic resonance imaging (MRI), data training, approximation theory, and portfolio selection. Since polynomial functions are non-convex, the problems under consideration are all NP-hard in general. In this paper we shall focus on polynomial-time approximation algorithms. In particular, we first study optimization of a multi-linear tensor function over the Cartesian product of spheres. We shall propose approximation algorithms for such problem and derive worst-case performance ratios, which are shown to be dependent only on the dimensions of the model. The methods are then extended to optimize a generic multi-variate homogeneous polynomial function with spherical constraint. Likewise, approximation algorithms are proposed with provable approximation performance ratios. Furthermore, the constraint set is relaxed to be an intersection of co-centered ellipsoids;namely, we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin, and propose polynomial-time approximation algorithms with provable worst-case performance ratios. Numerical results are reported, illustrating the effectiveness of the approximation algorithms studied.

关键词： Multi-linear tensor form polynomial function optimization Approximation algorithm

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Approximation algorithms for homogeneous polynomial optimization with quadratic constraints

Approximation algorithms for homogeneous polynomial optimiza...

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20th International Symposium of Mathematical Programming (ISMP)

关键词： Multi-linear tensor form polynomial function optimization Approximation algorithm

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Sum-of-Squares Design Method for Four-Parameter Lag-Lead Compensator

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Tsinghua Science and Technology 2009年第6期14卷 732-734页

作者：李志恒尹胜超张毅李力 Tsinghua National Laboratory for Information Science and Technology(TNList) Department of AutomationTsinghua University

The four-parameter lag-lead compensator design has received much attention in the last two decades. However, most approaches have been either trial-and-error or only for special cases. This paper presents a non-trial-and-error design method for four-parameter lag-lead compensators. Here, the compensator design problem is formulated into a polynomial function optimization problem and solved by using the recently developed sum-of-squares （SOS） techniques. This result not only provides a useful design method but also shows the power of the SOS techniques.

关键词： lag-lead compensator polynomial function optimization sum of squares （SOS） semi-definite programming （SDP）

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