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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Veröffentlicht in:	Mathematical programming Jg. 125; H. 2; S. 353 - 383
Hauptverfasser:	He, Simai, Li, Zhening, Zhang, Shuzhong
Format:	Journal Article Tagungsbericht
Sprache:	Englisch
Veröffentlicht:	Berlin/Heidelberg Springer-Verlag 01.10.2010 Springer Springer Nature B.V
Schlagworte:	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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Abstract	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.
AbstractList	Issue Title: 20th International Symposium on Mathematical Programming - ISMP 2009 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.[PUBLICATION ABSTRACT] 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.
Author	He, Simai Li, Zhening Zhang, Shuzhong
Author_xml	– sequence: 1 givenname: Simai surname: He fullname: He, Simai organization: Department of Management Sciences, City University of Hong Kong – sequence: 2 givenname: Zhening surname: Li fullname: Li, Zhening organization: Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong – sequence: 3 givenname: Shuzhong surname: Zhang fullname: Zhang, Shuzhong email: zhang@se.cuhk.edu.hk organization: Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong
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Keywords	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
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Snippet	In this paper, we consider approximation algorithms for optimizing a generic multi-variate homogeneous polynomial function, subject to homogeneous quadratic... Issue Title: 20th International Symposium on Mathematical Programming - ISMP 2009 In this paper, we consider approximation algorithms for optimizing a generic...
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SubjectTerms	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
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Title	Approximation algorithms for homogeneous polynomial optimization with quadratic constraints
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