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Numerical Experiments with the Lagrange Multiplier and Conjugate Gradient Methods (ILMCGM)

Received: 21 December 2014     Accepted: 25 December 2014     Published: 6 January 2015
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Abstract

In this paper, we imbed Langrage Multiplier Method (LMM) in Conjugate Gradient Method (CGM), which enables Conjugate Gradient Method (CGM) to be employed for solving constrained optimization problems of either equality, inequality constraint or both. In the past, Langrage Multiplier Method has been used extensively to solve constrained optimization problems. However, with some special features in CGM which makes it unique in solving unconstrained optimization problems, we see that this features we be advantageous to solve constrained optimization problems if we can add or subtract one or two things into the CGM. This, then call for the Numerical Experiments with the Lagrange Multiplier Conjugate Gradient Method (ILMCGM) that is aimed at taking care of any constrained optimization problems, either with equality or inequality constraint The authors of this paper desire that, with the construction of the Algorithm, one will circumvent the difficulties undergone using only LMM to solve constrained optimization problems and its application will further improve the result of the Conjugate Gradient Method in solving this class of optimization problem. We applied the new algorithm to some constrained optimization problems of two, three and four variables in which some of the problems are pertain to quadratic functions. Some of these functions are subject to linear, nonlinear, equality and inequality constraints.

Published in American Journal of Applied Mathematics (Volume 2, Issue 6)
DOI 10.11648/j.ajam.20140206.15
Page(s) 221-226
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Lagrange Multiplier Method, Constrained Optimization Problem, Conjugate Gradient Method, Numerical Experiments of the Lagrange Multiplier Conjugate Gradient Method

References
[1] RAO, S. S., (1978), Optimization Theory and Applications, Willy and Sons.
[2] THOMAS, F.E., and DAVID, M.H., (2001), Optimization of Chemical Processes, McGraw Hill
[3] IGOR, G., STEPHEN, G. N. and ARIELA, S., (2009), Linear and Nonlinear Optimization, George Mason University, Fairfax, Virginia, SIAM, Philadelphia.
[4] David, G. Hull, (2003), Optimal Control Theory for Applications, Mechanical Engineering Series, Springer-Verlag, New York, Inc., 175 Fifth Avenue, New York, NY 10010.
[5] Bersekas, D. P, (1982), Constrained Optimization and Lagrange Multipliers Method, Academic Press, Inc.
[6] Rockafellar, R. T., (2005), Multiplier Method of Hestenes and Powell applied to convex Programming, Journal of Optimization Theory and Applications, Vol. 4, No. 4.
[7] Triphath S. S and Narenda K. S, (1972), Constrained Optimization Problems Using Multiplier Methods, Journal of Optimization Theory and Applications: Vol. 9, No. 1.
Cite This Article
  • APA Style

    Samson Adebayo Olorunsola, Temitayo Emmanuel Olaosebikan, Kayode James Adebayo. (2015). Numerical Experiments with the Lagrange Multiplier and Conjugate Gradient Methods (ILMCGM). American Journal of Applied Mathematics, 2(6), 221-226. https://doi.org/10.11648/j.ajam.20140206.15

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    ACS Style

    Samson Adebayo Olorunsola; Temitayo Emmanuel Olaosebikan; Kayode James Adebayo. Numerical Experiments with the Lagrange Multiplier and Conjugate Gradient Methods (ILMCGM). Am. J. Appl. Math. 2015, 2(6), 221-226. doi: 10.11648/j.ajam.20140206.15

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    AMA Style

    Samson Adebayo Olorunsola, Temitayo Emmanuel Olaosebikan, Kayode James Adebayo. Numerical Experiments with the Lagrange Multiplier and Conjugate Gradient Methods (ILMCGM). Am J Appl Math. 2015;2(6):221-226. doi: 10.11648/j.ajam.20140206.15

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  • @article{10.11648/j.ajam.20140206.15,
      author = {Samson Adebayo Olorunsola and Temitayo Emmanuel Olaosebikan and Kayode James Adebayo},
      title = {Numerical Experiments with the Lagrange Multiplier and Conjugate Gradient Methods (ILMCGM)},
      journal = {American Journal of Applied Mathematics},
      volume = {2},
      number = {6},
      pages = {221-226},
      doi = {10.11648/j.ajam.20140206.15},
      url = {https://doi.org/10.11648/j.ajam.20140206.15},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20140206.15},
      abstract = {In this paper, we imbed Langrage Multiplier Method (LMM) in Conjugate Gradient Method (CGM), which enables Conjugate Gradient Method (CGM) to be employed for solving constrained optimization problems of either equality, inequality constraint or both. In the past, Langrage Multiplier Method has been used extensively to solve constrained optimization problems. However, with some special features in CGM which makes it unique in solving unconstrained optimization problems, we see that this features we be advantageous to solve constrained optimization problems if we can add or subtract one or two things into the CGM. This, then call for the Numerical Experiments with the Lagrange Multiplier Conjugate Gradient Method (ILMCGM) that is aimed at taking care of any constrained optimization problems, either with equality or inequality constraint The authors of this paper desire that, with the construction of the Algorithm, one will circumvent the difficulties undergone using only LMM to solve constrained optimization problems and its application will further improve the result of the Conjugate Gradient Method in solving this class of optimization problem. We applied the new algorithm to some constrained optimization problems of two, three and four variables in which some of the problems are pertain to quadratic functions. Some of these functions are subject to linear, nonlinear, equality and inequality constraints.},
     year = {2015}
    }
    

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    T1  - Numerical Experiments with the Lagrange Multiplier and Conjugate Gradient Methods (ILMCGM)
    AU  - Samson Adebayo Olorunsola
    AU  - Temitayo Emmanuel Olaosebikan
    AU  - Kayode James Adebayo
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    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
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    EP  - 226
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    UR  - https://doi.org/10.11648/j.ajam.20140206.15
    AB  - In this paper, we imbed Langrage Multiplier Method (LMM) in Conjugate Gradient Method (CGM), which enables Conjugate Gradient Method (CGM) to be employed for solving constrained optimization problems of either equality, inequality constraint or both. In the past, Langrage Multiplier Method has been used extensively to solve constrained optimization problems. However, with some special features in CGM which makes it unique in solving unconstrained optimization problems, we see that this features we be advantageous to solve constrained optimization problems if we can add or subtract one or two things into the CGM. This, then call for the Numerical Experiments with the Lagrange Multiplier Conjugate Gradient Method (ILMCGM) that is aimed at taking care of any constrained optimization problems, either with equality or inequality constraint The authors of this paper desire that, with the construction of the Algorithm, one will circumvent the difficulties undergone using only LMM to solve constrained optimization problems and its application will further improve the result of the Conjugate Gradient Method in solving this class of optimization problem. We applied the new algorithm to some constrained optimization problems of two, three and four variables in which some of the problems are pertain to quadratic functions. Some of these functions are subject to linear, nonlinear, equality and inequality constraints.
    VL  - 2
    IS  - 6
    ER  - 

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Author Information
  • Department of Mathematical Sciences, Ekiti State University, Ado Ekiti, Nigeria

  • Department of Mathematical Sciences, Ekiti State University, Ado Ekiti, Nigeria

  • Department of Mathematical Sciences, Ekiti State University, Ado Ekiti, Nigeria

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