American Institute of Mathematical Sciences

Advanced
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

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 & 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, (1983).   Google Scholar

[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.  doi: 10.1016/S0895-7177(03)90007-0.  Google Scholar

[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.   Google Scholar

[4]

M. Delgado Pineda and M. J. Muñoz Bouzo, "Lenguaje Matemático, Conjuntos y Números,", Sanz y Torres, (2011).   Google Scholar

[5]

E. A. Galperin, "The Cubic Algorithm for Optimization and Control,", NP Research Publ., (1990).   Google Scholar

[6]

E. A. Galperin, Set contraction algorithm for computing Pareto set in nonconvex nonsmooth multiobjective optimization,, Mathematical and Computer Modelling, 40 (2004), 847.  doi: 10.1016/j.mcm.2004.10.014.  Google Scholar

[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, (1979).   Google Scholar

[8]

V. Pareto, "Cours d'Êconomie Politique,", Lausanne Rouge, (1896).   Google Scholar

show all references

References:
[1]

V. Chankong and Y. Y. Haimes, "Multiobjective Decision Making, Theory and Methodology,", North-Holland, (1983).   Google Scholar

[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.  doi: 10.1016/S0895-7177(03)90007-0.  Google Scholar

[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.   Google Scholar

[4]

M. Delgado Pineda and M. J. Muñoz Bouzo, "Lenguaje Matemático, Conjuntos y Números,", Sanz y Torres, (2011).   Google Scholar

[5]

E. A. Galperin, "The Cubic Algorithm for Optimization and Control,", NP Research Publ., (1990).   Google Scholar

[6]

E. A. Galperin, Set contraction algorithm for computing Pareto set in nonconvex nonsmooth multiobjective optimization,, Mathematical and Computer Modelling, 40 (2004), 847.  doi: 10.1016/j.mcm.2004.10.014.  Google Scholar

[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, (1979).   Google Scholar

[8]

V. Pareto, "Cours d'Êconomie Politique,", Lausanne Rouge, (1896).   Google Scholar

[1]

Yong Wang, Wanquan Liu, Guanglu Zhou. An efficient algorithm for non-convex sparse optimization. Journal of Industrial & Management Optimization, 2019, 15 (4) : 2009-2021. doi: 10.3934/jimo.2018134
[2]

Tran Ngoc Thang, Nguyen Thi Bach Kim. Outcome space algorithm for generalized multiplicative problems and optimization over the efficient set. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1417-1433. doi: 10.3934/jimo.2016.12.1417
[3]

Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019102
[4]

Xiangyu Gao, Yong Sun. A new heuristic algorithm for laser antimissile strategy optimization. Journal of Industrial & Management Optimization, 2012, 8 (2) : 457-468. doi: 10.3934/jimo.2012.8.457
[5]

Adil Bagirov, Sona Taheri, Soodabeh Asadi. A difference of convex optimization algorithm for piecewise linear regression. Journal of Industrial & Management Optimization, 2019, 15 (2) : 909-932. doi: 10.3934/jimo.2018077
[6]

Jianjun Liu, Min Zeng, Yifan Ge, Changzhi Wu, Xiangyu Wang. Improved Cuckoo Search algorithm for numerical function optimization. Journal of Industrial & Management Optimization, 2020, 16 (1) : 103-115. doi: 10.3934/jimo.2018142
[7]

Cristian Barbarosie, Anca-Maria Toader, Sérgio Lopes. A gradient-type algorithm for constrained optimization with application to microstructure optimization. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1729-1755. doi: 10.3934/dcdsb.2019249
[8]

Henri Bonnel, Ngoc Sang Pham. Nonsmooth optimization over the (weakly or properly) Pareto set of a linear-quadratic multi-objective control problem: Explicit optimality conditions. Journal of Industrial & Management Optimization, 2011, 7 (4) : 789-809. doi: 10.3934/jimo.2011.7.789
[9]

Ping-Chen Lin. Portfolio optimization and risk measurement based on non-dominated sorting genetic algorithm. Journal of Industrial & Management Optimization, 2012, 8 (3) : 549-564. doi: 10.3934/jimo.2012.8.549
[10]

Jiawei Chen, Guangmin Wang, Xiaoqing Ou, Wenyan Zhang. Continuity of solutions mappings of parametric set optimization problems. Journal of Industrial & Management Optimization, 2020, 16 (1) : 25-36. doi: 10.3934/jimo.2018138
[11]

Hong-Zhi Wei, Chun-Rong Chen. Three concepts of robust efficiency for uncertain multiobjective optimization problems via set order relations. Journal of Industrial & Management Optimization, 2019, 15 (2) : 705-721. doi: 10.3934/jimo.2018066
[12]

Nguyen Van Thoai. Decomposition branch and bound algorithm for optimization problems over efficient sets. Journal of Industrial & Management Optimization, 2008, 4 (4) : 647-660. doi: 10.3934/jimo.2008.4.647
[13]

Lianshuan Shi, Enmin Feng, Huanchun Sun, Zhaosheng Feng. A two-step algorithm for layout optimization of structures with discrete variables. Journal of Industrial & Management Optimization, 2007, 3 (3) : 543-552. doi: 10.3934/jimo.2007.3.543
[14]

Lipu Zhang, Yinghong Xu, Zhengjing Jin. An efficient algorithm for convex quadratic semi-definite optimization. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 129-144. doi: 10.3934/naco.2012.2.129
[15]

Behrouz Kheirfam, Morteza Moslemi. On the extension of an arc-search interior-point algorithm for semidefinite optimization. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 261-275. doi: 10.3934/naco.2018015
[16]

Paul B. Hermanns, Nguyen Van Thoai. Global optimization algorithm for solving bilevel programming problems with quadratic lower levels. Journal of Industrial & Management Optimization, 2010, 6 (1) : 177-196. doi: 10.3934/jimo.2010.6.177
[17]

Chunlin Hao, Xinwei Liu. Global convergence of an SQP algorithm for nonlinear optimization with overdetermined constraints. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 19-29. doi: 10.3934/naco.2012.2.19
[18]

Chia-Huang Wu, Kuo-Hsiung Wang, Jau-Chuan Ke, Jyh-Bin Ke. A heuristic algorithm for the optimization of M/M/$s$ queue with multiple working vacations. Journal of Industrial & Management Optimization, 2012, 8 (1) : 1-17. doi: 10.3934/jimo.2012.8.1
[19]

Xin Zhang, Jie Wen, Qin Ni. Subspace trust-region algorithm with conic model for unconstrained optimization. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 223-234. doi: 10.3934/naco.2013.3.223
[20]

Soheil Dolatabadi. Weighted vertices optimizer (WVO): A novel metaheuristic optimization algorithm. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 461-479. doi: 10.3934/naco.2018029

 Impact Factor: 

Tools

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences