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Constrained Optimization

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Optimization Techniques in Computer Vision

Part of the book series: Advances in Computer Vision and Pattern Recognition ((ACVPR))

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Abstract

In most practical image processing and computer vision problems involving optimization, the solutions do not span the entire range of real values. This resulting bounding property is often imposed upon the solution process in the form of constraints placed upon system parameters. Optimization problems with constraints are generally referred to as constrained optimization problems.

The basic idea of solving most constrained optimization problems is to determine the solution that satisfies the stated performance index, subject to the given constraints. In this chapter, proper methods of finding the solutions to constrained optimization problems, according to the characteristics of various types of constraints, are discussed.

If the solution and its local perturbation do not violate the given constraints, we consider the problem as being unconstrained. Therefore, the important issue of constrained optimization is analyzing the local behavior of the solution on or near the boundary of constraints.

This chapter includes three sections. The first section describes the constrained optimization problem and defines the terminology commonly used with this problem, both in the literature and in this chapter. The second and the third sections describe optimality conditions and the corresponding optimization methods with linear and nonlinear constraints, respectively.

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Notes

  1. 1.

    We note that in the single linear equation constraint case, the projected gradient is a scalar quantity, while in the multiple linear equations constraint case, it is an \( M\times 1 \) vector, where M represents the number of constraints.

References and Further Readings

  1. D.P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods (Academic, New York, 1982)

    MATH  Google Scholar 

  2. P.E. Gill, W. Murray, M.A. Saunders, M.H. Wright, Constrained nonlinear programming, in Optimization, ed. by G.L. Nemhauser, A.H.G. Rinnooy Kan, M.J. Todd (North-Holland, Amsterdam, 1989), pp. 171–210

    Chapter  Google Scholar 

  3. J. Goldsmith, A.H. Barr, Applying constrained optimization to computer graphics. SMPTE J. 102(10), 910–912 (1993)

    Article  Google Scholar 

  4. P.M. Pardalos, J.B. Rosen, Constrained Global Optimization: Algorithms and Applications (Springer, New York, 1987)

    Book  MATH  Google Scholar 

  5. C.A. Floudas, P.M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms. Lecture Notes in Computer Science (Springer, Berlin, 1990)

    Book  MATH  Google Scholar 

  6. P.E. Gill, W. Murray, M.A. Saunders, M.H. Wright, Linearly constrained optimization, in Nonlinear Optimization 1981, ed. by M.J.D. Powell (Academic Press, London, 1982), pp. 123–139

    Google Scholar 

  7. P.E. Gill, W. Murray, M.A. Saunders, M.H. Wright, Recent developments in constrained optimization. J. Comput. Appl. Math. 22, 257–270 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  8. P.E. Gill, W. Murray, M.A. Saunders, M.H. Wright, Constrained nonlinear programming, in Optimization, ed. by G.L. Nemhauser, A.H.G. Rinnooy Kan, M.J. Todd (Elsevier, Amsterdam, 1989), pp. 171–210

    Chapter  Google Scholar 

  9. P.E. Gill, W. Murray, M.A. Saunders, M.H. Wright, A practical anti-cycling procedure for linearly constrained optimization. Math. Program. 45, 437–474 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. M.H. Wright, Interior methods for constrained optimization, in Acta Numerica, vol 1 (Cambridge University Press, Cambridge, 1992), pp. 341–407

    Google Scholar 

  11. M.H. Wright, The interior-point revolution in constrained optimization, in High-Performance Algorithms and Software in Nonlinear Optimization, ed. by R. DeLeone, A. Murli, P.M. Pardalos, G. Toraldo (Kluwer Academic Publishers, Dordrecht, 1998), pp. 359–381

    Chapter  Google Scholar 

  12. A.R. Conn, N.I.M. Gould, P.L. Toint, Large-scale optimization. Math. Program. B 45, 3 (1989)

    MATH  Google Scholar 

  13. A.R. Conn, N.I.M. Gould, P.L. Toint, Large-scale optimization—applications. Math. Program. B 48, 1 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gert Smolka, Problem solving with constraints and programming. ACM Comput. Surv. 28(4es) (December 1996). Electronic Section

    Google Scholar 

  15. R.P. Brent, Algorithms for Minimization Without Derivatives (Prentice Hall, Englewood Cliffs, NJ, 1973)

    MATH  Google Scholar 

  16. E. Chong, S. Zak, An Introduction to Optimization (Wiley, New York, 1996)

    MATH  Google Scholar 

  17. S.C. Fang, S. Puthenpura, Linear Optimization and Extensions (Prentice Hall, Englewood Cliffs, NJ, 1993)

    MATH  Google Scholar 

  18. R.W. Farebrother, Linear Least Squares Computation (Marcel Dekker, New York, 1987)

    MATH  Google Scholar 

  19. G.H. Golub, Numerical methods for solving linear least squares problems. Numer. Math. 7, 206–216 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  20. G. Strang, Introduction to Applied Mathematics (Wellesley-Cambridge, Wellesley, MA, 1986)

    MATH  Google Scholar 

  21. G. Strang, Linear Algebra and Its Applications, 3rd edn. (W. B. Saunders, New York, 1988)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Tennessee, Knoxville, Tennessee, USA

    Mongi A. Abidi

  2. Department of Human Factors, Controls, and Statistics, Idaho National Laboratory, Idaho Falls, Idaho, USA

    Andrei V. Gribok

  3. Image Processing and Intelligent Systems Laboratory, Chung-Ang University, Seoul, Korea

    Joonki Paik

Authors
  1. Mongi A. Abidi
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  2. Andrei V. Gribok
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  3. Joonki Paik
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Abidi, M.A., Gribok, A.V., Paik, J. (2016). Constrained Optimization. In: Optimization Techniques in Computer Vision. Advances in Computer Vision and Pattern Recognition. Springer, Cham. https://doi.org/10.1007/978-3-319-46364-3_5

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  • DOI: https://doi.org/10.1007/978-3-319-46364-3_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-46363-6

  • Online ISBN: 978-3-319-46364-3

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