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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Vydané v:Mathematical programming Ročník 125; číslo 2; s. 353 - 383
Hlavní autori: He, Simai, Li, Zhening, Zhang, Shuzhong
Médium: Journal Article Konferenčný príspevok..
Jazyk:English
Vydavateľské údaje: Berlin/Heidelberg Springer-Verlag 01.10.2010
Springer
Springer Nature B.V
Predmet:
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.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-010-0409-z