Approximation algorithms for homogeneous polynomial optimization with quadratic constraints

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, appr...

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Vydáno v:	Mathematical programming Ročník 125; číslo 2; s. 353 - 383
Hlavní autoři:	He, Simai, Li, Zhening, Zhang, Shuzhong
Médium:	Journal Article Konferenční příspěvek
Jazyk:	angličtina
Vydáno:	Berlin/Heidelberg Springer-Verlag 01.10.2010 Springer Springer Nature B.V
Témata:	Algorithms Applied sciences Approximation Calculus of Variations and Optimal Control; Optimization Combinatorics Ellipsoids Exact sciences and technology Full Length Paper Intersections Linear algebra Magnetic resonance imaging Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical models Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Maximization Numerical Analysis Operational research and scientific management Operational research. Management science Optimization Polynomials Quantum physics Signal processing Studies Theoretical Voice recognition Multi-linear tensor form 90C26 90C59 15A69 Approximation algorithm Polynomial function optimization Intersection Non convex programming Linear form Polynomial approximation Worst case method Constraint satisfaction Polynomial method Modeling Convex programming Sphere Portfolio selection Convex function Approximation theory Cartesian product Mathematical programming Quadratic programming Set constraint Nuclear magnetic resonance imaging Constrained optimization Polynomial time Polynomial function NP hard problem Signal processing Homogeneous function Ellipsoid Portfolio management Non convex analysis
ISSN:	0025-5610, 1436-4646
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Shrnutí:	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.
Bibliografie:	SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-1 ObjectType-Feature-2 content type line 23
ISSN:	0025-5610 1436-4646
DOI:	10.1007/s10107-010-0409-z