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Analytical solutions to the optimization of a quadratic cost function subject to linear and quadratic equality constraints

Analytical solutions to the optimization of a quadratic cost function subject to linear and... In the area of broad-band antenna array signal processing, the global minimum of a quadratic equality constrained quadratic cost minimization problem is often required. The problem posed is usually characterized by a large optimization space (around 50–90 tuples), a large number of linear equality constraints, and a few quadratic equality constraints each having very low rank quadratic constraint matrices. Two main difficulties arise in this class of problem. Firstly, the feasibility region is nonconvex and multiple local minima abound. This makes conventional numerical search techniques unattractive as they are unable to locate the global optimum consistently (unless a finite search area is specified). Secondly, the large optimization space makes the use of decision-method algorithms for the theory of the reals unattractive. This is because these algorithms involve the solution of the roots of univariate polynomials of order to the square of the optimization space. In this paper we present a new algorithm which exploits the structure of the constraints to reduce the optimization space to a more manageable size. The new algorithm relies on linear-algebra concepts, basic optimization theory, and a multivariate polynomial root-solving tool often used by decision-method algorithms. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Analytical solutions to the optimization of a quadratic cost function subject to linear and quadratic equality constraints

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Publisher
Springer Journals
Copyright
Copyright © 1996 by Springer-Verlag New York Inc.
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
DOI
10.1007/BF01182622
Publisher site
See Article on Publisher Site

Abstract

In the area of broad-band antenna array signal processing, the global minimum of a quadratic equality constrained quadratic cost minimization problem is often required. The problem posed is usually characterized by a large optimization space (around 50–90 tuples), a large number of linear equality constraints, and a few quadratic equality constraints each having very low rank quadratic constraint matrices. Two main difficulties arise in this class of problem. Firstly, the feasibility region is nonconvex and multiple local minima abound. This makes conventional numerical search techniques unattractive as they are unable to locate the global optimum consistently (unless a finite search area is specified). Secondly, the large optimization space makes the use of decision-method algorithms for the theory of the reals unattractive. This is because these algorithms involve the solution of the roots of univariate polynomials of order to the square of the optimization space. In this paper we present a new algorithm which exploits the structure of the constraints to reduce the optimization space to a more manageable size. The new algorithm relies on linear-algebra concepts, basic optimization theory, and a multivariate polynomial root-solving tool often used by decision-method algorithms.

Journal

Applied Mathematics and Optimization – Springer Journals

Published: Feb 2, 2005

References

  • Adaptive Arrays with Main Beam Constraints
    Applebaum, S. P.; Chapman, D. J.
  • Multiple-Constraint Adaptive Filtering
    Booker, A.; Ong, C. Y.
  • Applied Optimal Control
    Bryson, A. E.; Ho, Y. C.
  • Symbolic and Algebraic Computation by Computers
    Buchberger, B.
  • Multidimensional Systems Theory
    Buchberger, B.
  • Proc. IEEE Symp.
    Canny, J.
  • Generalized Characteristic Polynomials
    Canny, J. F.
  • Resolving Powers and Sensitivity to Mismatch of Optimum Array Processors
    Cox, H.
  • Techniques in Robust Broad Band Beamforming, Control and Dynamic Systems
    Er, M. H.; Cantoni, A.
  • Practical Methods of Optimization
    Fletcher, R.
  • An Algorithm for Linearly Constrained Adaptive Antenna Array Processing
    Frost, O. L.
  • Practical Optimization
    Gill, P. E.; Murray, W.; Wright, M. H.
  • Numerical Linear Algebra and Optimization, vol. 1
    Gill, P. E.; Murray, W.; Wright, M. H.
  • Quiescent Pattern Control in Linearly Constrained Adaptive Arrays
    Griffiths, L. J.; Buckley, K. M.
  • Global Optimization
    Horst, R.; Tuy, Hoang
  • Adaptive Combining of Wideband Array Data for Optimal Reception
    Lacoss, R. T.
  • Linear and Non-linear Programming
    Luenberger, D. G.
  • Lecture Notes in Computer Science, vol. 268
    Pardalos, P. M.; Rosen, J. B.
  • On the Computational Complexity of Approximating Solutions for Real Algebraic Formulae
    Renegar, J.
  • On the Computational Complexity and Geometry of the First Order Theory of the Reals. Part I: Introduction: Preliminaries; the Geometry of Semi-Algebraic Sets; the Decision Problem for the Existential Theory of the Reals
    Renegar, J.
  • On the Computational Complexity and Geometry of the First Order Theory of the Reals. Part II: the General Decision Problem; Preliminaries for Quantifier Elimination
    Renegar, J.
  • An Adaptive Antenna Array under Directional Constraint
    Takao, K.; Fujita, M.; Nishi, T.
  • Derivative Constrained Optimum Broad Band Antenna Arrays
    Thng, I.; Cantoni, A.; Leung, Y. H.
  • Lecture Notes in Computer Science, vol. 350
    Torn, A.; Zilinskas, A.
  • Minimum Variance Beamforming with Phase-Independent Derivative Constraints
    Tseng, C. Y.
  • Modern Algebra, vol. 2
    Waerden, B. L.
  • Rounding Errors in Algebraic Processes
    Wilkinson, J. H.