MAPLE code of the cubic algorithm for multiobjective optimization with box constraints

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2013, 3(3): 407-424. doi: 10.3934/naco.2013.3.407

MAPLE code of the cubic algorithm for multiobjective optimization with box constraints

M. Delgado Pineda ^1, , E. A. Galperin ^2, and P. Jiménez Guerra ^1,

1.	Departamento de Matemáticas Fundamentales, Facultad de Ciencias, Universidad Nacional de Educación a Distancia, UNED, c/ Senda del Rey 9, C.P. 28040 Madrid, Spain, Spain
2.	Département de mathématiques, Université du Québec à Montréal, C.P. 8888, Succ. Centre Ville, Montréal, Québec H3C 3P8, Canada

Received December 2011 Revised February 2013 Published July 2013

Abstract
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A generalization of the cubic algorithm is presented for global optimization of nonconvex nonsmooth multiobjective optimization programs $\min f_{s}(x),\ s=1,\dots,k,$ with box constraints $x\in X=[a_{1},b_{1}]\times \dots\times\lbrack a_{n},b_{n}]$.
This monotonic set contraction algorithm converges onto the entire exact Pareto set, if nonempty, and yields its approximation with given precision in a finite number of iterations. Simultaneously, approximations for the ideal point and for the function values over Pareto set are obtained. The method is implemented by Maple code, and this code does not create ill-conditioned situations.
Results of numerical experiments are presented, with graphs, to illustrate the use of the code, and the solution set can be visualized in projections on coordinate planes. The code is ready for engineering and economic applications.

Keywords: non-differentiable optimization., cubic algorithm, Multiobjective optimization, set contraction algorithm, Pareto solutions.

Mathematics Subject Classification: Primary: 90C26; Secondary: 58E1.

Citation: M. Delgado Pineda, E. A. Galperin, P. Jiménez Guerra. MAPLE code of the cubic algorithm for multiobjective optimization with box constraints. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 407-424. doi: 10.3934/naco.2013.3.407

References:

[1]	V. Chankong and Y. Y. Haimes, "Multiobjective Decision Making, Theory and Methodology," North-Holland, New York, 1983.
[2]	M. Delgado Pineda and E. A. Galperin, Global optimization in Rⁿ with box constraints and applications: A maple code, Mathematical and Computer Modelling, 38 (2003), 77-97. doi: 10.1016/S0895-7177(03)90007-0.
[3]	M. Delgado Pineda and E. A. Galperin, Global optimization over general compact sets by the Beta algorithm: A maple code, Computer and Mathematics with Applications, 52 (2006), 33-54.
[4]	M. Delgado Pineda and M. J. Muñoz Bouzo, "Lenguaje Matemático, Conjuntos y Números," Sanz y Torres, Madrid, 2011.
[5]	E. A. Galperin, "The Cubic Algorithm for Optimization and Control," NP Research Publ., Montreal, 1990.
[6]	E. A. Galperin, Set contraction algorithm for computing Pareto set in nonconvex nonsmooth multiobjective optimization, Mathematical and Computer Modelling, 40 (2004), 847-859. doi: 10.1016/j.mcm.2004.10.014.
[7]	C. L. Hwang, A. S .M. Masud, S. R. Paidy and K. Yoon, "Multiple Objectives Decision Making, Methods and Applications: A State of the Art Survey," Springer-Verlang, Berlin, 1979.
[8]	V. Pareto, "Cours d'Êconomie Politique," Lausanne Rouge, Lausanne, Switzerland, 1896.

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References:

[1]	V. Chankong and Y. Y. Haimes, "Multiobjective Decision Making, Theory and Methodology," North-Holland, New York, 1983.
[2]	M. Delgado Pineda and E. A. Galperin, Global optimization in Rⁿ with box constraints and applications: A maple code, Mathematical and Computer Modelling, 38 (2003), 77-97. doi: 10.1016/S0895-7177(03)90007-0.
[3]	M. Delgado Pineda and E. A. Galperin, Global optimization over general compact sets by the Beta algorithm: A maple code, Computer and Mathematics with Applications, 52 (2006), 33-54.
[4]	M. Delgado Pineda and M. J. Muñoz Bouzo, "Lenguaje Matemático, Conjuntos y Números," Sanz y Torres, Madrid, 2011.
[5]	E. A. Galperin, "The Cubic Algorithm for Optimization and Control," NP Research Publ., Montreal, 1990.
[6]	E. A. Galperin, Set contraction algorithm for computing Pareto set in nonconvex nonsmooth multiobjective optimization, Mathematical and Computer Modelling, 40 (2004), 847-859. doi: 10.1016/j.mcm.2004.10.014.
[7]	C. L. Hwang, A. S .M. Masud, S. R. Paidy and K. Yoon, "Multiple Objectives Decision Making, Methods and Applications: A State of the Art Survey," Springer-Verlang, Berlin, 1979.
[8]	V. Pareto, "Cours d'Êconomie Politique," Lausanne Rouge, Lausanne, Switzerland, 1896.

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