A New Approach for Kuhn-Tucker Conditions to Solve Quadratic Programming Problems with Linear Inequality Constraints

Ayansola Olufemi Aderemi; Adejumo Adebowale Olusola

doi:doi:10.11648/j.mcs.20200505.11

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A New Approach for Kuhn-Tucker Conditions to Solve Quadratic Programming Problems with Linear Inequality Constraints

Ayansola Olufemi Aderemi, Adejumo Adebowale Olusola

Published in Mathematics and Computer Science (Volume 5, Issue 5)

Received: 27 May 2020 Accepted: 18 September 2020 Published: 11 December 2020

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Abstract

Many real-life problems, such as economic, industrial, engineering to mention but a few has been dealt with, using linear programming that assumes linearity in the objective function and constraint functions. It is noteworthy that there are many situations where the objective function and / or some or all of the constraints are non-linear functions. It is observed that many researchers have laboured so much at finding general solution approach to Non-linear programming problems but all to no avail. Of the prominent methods of solution of Non-linear programming problems: Karush- Kuhn-Tucker conditions method and Wolf modified simplex method. The Karush-Kuhn-Tucker theorem gives necessary and sufficient conditions for the existence of an optimal solution to non-linear programming problems, a finite-dimensional optimization problem where the variables have to fulfill some inequality constraints while Wolf in addition to Karush- Kuhn-Tucker conditions, modified the simplex method after changing quadratic linear function in the objective function to linear function. In this paper, an alternative method for Karush-Kuhn-Tucker conditional method is proposed. This method is simpler than the two methods considered to solve quadratic programming problems of maximizing quadratic objective function subject to a set of linear inequality constraints. This is established because of its computational efforts.

Published in	Mathematics and Computer Science (Volume 5, Issue 5)
DOI	10.11648/j.mcs.20200505.11
Page(s)	86-92
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), 2024. Published by Science Publishing Group

Keywords

Karush- Kuhn-Tucker Conditions, New Approach, Quadratic Programming, Wolf Modified Simplex Method

References

[1]	Kuhn, H. W., and Tucker, A. W. (1950) Non-linear Programming In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, J. Neyman, Ed., pp. 481-492, Berkeley.
[2]	Karush, W. (1939) Minima of Functions of Several Variables with Inequalities as Side Conditions. Dissertation, Department of Mathematics, University of Chicago.
[3]	Wolfe Philip (1959) The Simplex Method for Quadratic Programming, The Econometric Society, Econometrica, Vol. 27, pp. 382-398.
[4]	Allran, R. R and Johnsen, S. E. J (1970) “An Algorithm for Solving Non-linear programming problems subject to non-linear inequality constraints” The computer Journal, Vol. 13, No 2. Pp 171-177.
[5]	Terlaky T. A. (1987) New Algorithm for Quadratic Programming, European Journal of Operation Research, Vol. 32, pp. 294-301, North-Holland.
[6]	Frank M. and Wolfe, P. (1956) An Algorithm for Quadratic Programming, Naval Research Logistics Quarterly March-June, pp. 95-110.
[7]	Pawar, T. S and Ghadle, K. P. (2015) “New approach for Wolfe’s modified simplex method to solve Quadratic programming problems” International Journal of Research in Engineering and Technology, vol. 04, issue 01, pp. 371-376.
[8]	Zoufendijk, G. (1960) Methods of Feasible Directions. Amsterdem and New York: Elsevier Publishing Company.
[9]	Rosen, J. B (1961) The Gradient Project Method for Nonlinear Programming. Part II: Nonlinear Constraints, Shall Development Company, Emeryville, California.
[10]	Fiacco, A. V. and McCormick, G. P. (1963) Programming under Nonlinear Constraints by Unconstrained Minimization: A Primal-Dual Method, Research Analysis Corporation, McLean, Virginia, Technical Paper, RAC-TP-96.
[11]	Li, J. X. (1994). On an Algorithm for Solving Fuzzy Linear Systems, Fuzzy Sets and Systems 61, 369-371.
[12]	Tang, J. and Wang, D. (1996) Modelling and Optimization for a type of Fuzzy Nonlinear Programming Problems in Manufacturing Systems, Proceedings of the 35th Conference on Decision and Control, Kobe, Japan.
[13]	Iyengar, P. (2002) “Non-Linear Programming; Introduction”, IEOR, Handout 19, 16 October.
[14]	Song, Y. Chen, Y. and Wu, X. (2005) “A Method for Solving Nonlinear Programming Models with All Fuzzy Coefficients Based on Genetic Algorithm”, Advances in Natural Computation, Vol. 36, No. 11, pp, 1101-1104.
[15]	Nasseri, S. H. (2008) “Fuzzy Nonlinear Optimization, “The Journal of Nonlinear Analysis and its Applications, Vol. 1, No. 4, pp. 230-235.
[16]	Ali, F. J. and Amir, S. (2012). “Solving Nonlinear Programming Problem in Fuzzy Environment”, Int. J. Contemp. Math. Sciences, Vol. 7, No. 4, pp. 159-170.
[17]	Nayak, J. and Sanjaya, K. B. (2012). “Optimal Solution of Fuzzy Nonlinear Programming Problems with Linear Constraints”, International Journal of Advances in Science and Technology, Vol. 4, No. 4, pp. 43-52.

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Ayansola Olufemi Aderemi, Adejumo Adebowale Olusola. (2020). A New Approach for Kuhn-Tucker Conditions to Solve Quadratic Programming Problems with Linear Inequality Constraints. Mathematics and Computer Science, 5(5), 86-92. https://doi.org/10.11648/j.mcs.20200505.11

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ACS Style

Ayansola Olufemi Aderemi; Adejumo Adebowale Olusola. A New Approach for Kuhn-Tucker Conditions to Solve Quadratic Programming Problems with Linear Inequality Constraints. Math. Comput. Sci. 2020, 5(5), 86-92. doi: 10.11648/j.mcs.20200505.11

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AMA Style

Ayansola Olufemi Aderemi, Adejumo Adebowale Olusola. A New Approach for Kuhn-Tucker Conditions to Solve Quadratic Programming Problems with Linear Inequality Constraints. Math Comput Sci. 2020;5(5):86-92. doi: 10.11648/j.mcs.20200505.11

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@article{10.11648/j.mcs.20200505.11,
  author = {Ayansola Olufemi Aderemi and Adejumo Adebowale Olusola},
  title = {A New Approach for Kuhn-Tucker Conditions to Solve Quadratic Programming Problems with Linear Inequality Constraints},
  journal = {Mathematics and Computer Science},
  volume = {5},
  number = {5},
  pages = {86-92},
  doi = {10.11648/j.mcs.20200505.11},
  url = {https://doi.org/10.11648/j.mcs.20200505.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20200505.11},
  abstract = {Many real-life problems, such as economic, industrial, engineering to mention but a few has been dealt with, using linear programming that assumes linearity in the objective function and constraint functions. It is noteworthy that there are many situations where the objective function and / or some or all of the constraints are non-linear functions. It is observed that many researchers have laboured so much at finding general solution approach to Non-linear programming problems but all to no avail. Of the prominent methods of solution of Non-linear programming problems: Karush- Kuhn-Tucker conditions method and Wolf modified simplex method. The Karush-Kuhn-Tucker theorem gives necessary and sufficient conditions for the existence of an optimal solution to non-linear programming problems, a finite-dimensional optimization problem where the variables have to fulfill some inequality constraints while Wolf in addition to Karush- Kuhn-Tucker conditions, modified the simplex method after changing quadratic linear function in the objective function to linear function. In this paper, an alternative method for Karush-Kuhn-Tucker conditional method is proposed. This method is simpler than the two methods considered to solve quadratic programming problems of maximizing quadratic objective function subject to a set of linear inequality constraints. This is established because of its computational efforts.},
 year = {2020}
}

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TY  - JOUR
T1  - A New Approach for Kuhn-Tucker Conditions to Solve Quadratic Programming Problems with Linear Inequality Constraints
AU  - Ayansola Olufemi Aderemi
AU  - Adejumo Adebowale Olusola
Y1  - 2020/12/11
PY  - 2020
N1  - https://doi.org/10.11648/j.mcs.20200505.11
DO  - 10.11648/j.mcs.20200505.11
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 86
EP  - 92
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20200505.11
AB  - Many real-life problems, such as economic, industrial, engineering to mention but a few has been dealt with, using linear programming that assumes linearity in the objective function and constraint functions. It is noteworthy that there are many situations where the objective function and / or some or all of the constraints are non-linear functions. It is observed that many researchers have laboured so much at finding general solution approach to Non-linear programming problems but all to no avail. Of the prominent methods of solution of Non-linear programming problems: Karush- Kuhn-Tucker conditions method and Wolf modified simplex method. The Karush-Kuhn-Tucker theorem gives necessary and sufficient conditions for the existence of an optimal solution to non-linear programming problems, a finite-dimensional optimization problem where the variables have to fulfill some inequality constraints while Wolf in addition to Karush- Kuhn-Tucker conditions, modified the simplex method after changing quadratic linear function in the objective function to linear function. In this paper, an alternative method for Karush-Kuhn-Tucker conditional method is proposed. This method is simpler than the two methods considered to solve quadratic programming problems of maximizing quadratic objective function subject to a set of linear inequality constraints. This is established because of its computational efforts.
VL  - 5
IS  - 5
ER  -

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Author Information

Ayansola Olufemi Aderemi

Department of Mathematics & Statistics, The Polytechnic Ibadan, Ibadan, Nigeria
Adejumo Adebowale Olusola

Department of Statistics, University of Ilorin, Ilorin, Nigeria

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APA Style

Ayansola Olufemi Aderemi, Adejumo Adebowale Olusola. (2020). A New Approach for Kuhn-Tucker Conditions to Solve Quadratic Programming Problems with Linear Inequality Constraints. Mathematics and Computer Science, 5(5), 86-92. https://doi.org/10.11648/j.mcs.20200505.11

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ACS Style

Ayansola Olufemi Aderemi; Adejumo Adebowale Olusola. A New Approach for Kuhn-Tucker Conditions to Solve Quadratic Programming Problems with Linear Inequality Constraints. Math. Comput. Sci. 2020, 5(5), 86-92. doi: 10.11648/j.mcs.20200505.11

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AMA Style

Ayansola Olufemi Aderemi, Adejumo Adebowale Olusola. A New Approach for Kuhn-Tucker Conditions to Solve Quadratic Programming Problems with Linear Inequality Constraints. Math Comput Sci. 2020;5(5):86-92. doi: 10.11648/j.mcs.20200505.11

Copy | Download

@article{10.11648/j.mcs.20200505.11,
  author = {Ayansola Olufemi Aderemi and Adejumo Adebowale Olusola},
  title = {A New Approach for Kuhn-Tucker Conditions to Solve Quadratic Programming Problems with Linear Inequality Constraints},
  journal = {Mathematics and Computer Science},
  volume = {5},
  number = {5},
  pages = {86-92},
  doi = {10.11648/j.mcs.20200505.11},
  url = {https://doi.org/10.11648/j.mcs.20200505.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20200505.11},
  abstract = {Many real-life problems, such as economic, industrial, engineering to mention but a few has been dealt with, using linear programming that assumes linearity in the objective function and constraint functions. It is noteworthy that there are many situations where the objective function and / or some or all of the constraints are non-linear functions. It is observed that many researchers have laboured so much at finding general solution approach to Non-linear programming problems but all to no avail. Of the prominent methods of solution of Non-linear programming problems: Karush- Kuhn-Tucker conditions method and Wolf modified simplex method. The Karush-Kuhn-Tucker theorem gives necessary and sufficient conditions for the existence of an optimal solution to non-linear programming problems, a finite-dimensional optimization problem where the variables have to fulfill some inequality constraints while Wolf in addition to Karush- Kuhn-Tucker conditions, modified the simplex method after changing quadratic linear function in the objective function to linear function. In this paper, an alternative method for Karush-Kuhn-Tucker conditional method is proposed. This method is simpler than the two methods considered to solve quadratic programming problems of maximizing quadratic objective function subject to a set of linear inequality constraints. This is established because of its computational efforts.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - A New Approach for Kuhn-Tucker Conditions to Solve Quadratic Programming Problems with Linear Inequality Constraints
AU  - Ayansola Olufemi Aderemi
AU  - Adejumo Adebowale Olusola
Y1  - 2020/12/11
PY  - 2020
N1  - https://doi.org/10.11648/j.mcs.20200505.11
DO  - 10.11648/j.mcs.20200505.11
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 86
EP  - 92
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20200505.11
AB  - Many real-life problems, such as economic, industrial, engineering to mention but a few has been dealt with, using linear programming that assumes linearity in the objective function and constraint functions. It is noteworthy that there are many situations where the objective function and / or some or all of the constraints are non-linear functions. It is observed that many researchers have laboured so much at finding general solution approach to Non-linear programming problems but all to no avail. Of the prominent methods of solution of Non-linear programming problems: Karush- Kuhn-Tucker conditions method and Wolf modified simplex method. The Karush-Kuhn-Tucker theorem gives necessary and sufficient conditions for the existence of an optimal solution to non-linear programming problems, a finite-dimensional optimization problem where the variables have to fulfill some inequality constraints while Wolf in addition to Karush- Kuhn-Tucker conditions, modified the simplex method after changing quadratic linear function in the objective function to linear function. In this paper, an alternative method for Karush-Kuhn-Tucker conditional method is proposed. This method is simpler than the two methods considered to solve quadratic programming problems of maximizing quadratic objective function subject to a set of linear inequality constraints. This is established because of its computational efforts.
VL  - 5
IS  - 5
ER  -

Copy | Download