A Penalty Function Algorithm with Objective Parameters and Constraint Penalty Parameter for Multi-Objective Programming
Affiliation(s)
College of Business and Administration, Zhejiang University of Technology, Hangzhou, China.
College of Business and Administration, Zhejiang University of Technology, Hangzhou, China.
ABSTRACT
In this paper, we present an algorithm to solve the inequality constrained multi-objective programming (MP) by using a penalty function with objective parameters and constraint penalty parameter. First, the penalty function with objective parameters and constraint penalty parameter for MP and the corresponding unconstraint penalty optimization problem (UPOP) is defined. Under some conditions, a Pareto efficient solution (or a weakly-efficient solution) to UPOP is proved to be a Pareto efficient solution (or a weakly-efficient solution) to MP. The penalty function is proved to be exact under a stable condition. Then, we design an algorithm to solve MP and prove its convergence. Finally, numerical examples show that the algorithm may help decision makers to find a satisfactory solution to MP.
In this paper, we present an algorithm to solve the inequality constrained multi-objective programming (MP) by using a penalty function with objective parameters and constraint penalty parameter. First, the penalty function with objective parameters and constraint penalty parameter for MP and the corresponding unconstraint penalty optimization problem (UPOP) is defined. Under some conditions, a Pareto efficient solution (or a weakly-efficient solution) to UPOP is proved to be a Pareto efficient solution (or a weakly-efficient solution) to MP. The penalty function is proved to be exact under a stable condition. Then, we design an algorithm to solve MP and prove its convergence. Finally, numerical examples show that the algorithm may help decision makers to find a satisfactory solution to MP.
KEYWORDS
Multi-Objective Programming, Penalty Function, Objective Parameters, Constraint Penalty Parameter, Pareto Weakly-Efficient Solution
Multi-Objective Programming, Penalty Function, Objective Parameters, Constraint Penalty Parameter, Pareto Weakly-Efficient Solution
Cite this paper
Meng, Z. , Shen, R. and Jiang, M. (2014) A Penalty Function Algorithm with Objective Parameters and Constraint Penalty Parameter for Multi-Objective Programming. American Journal of Operations Research, 4, 331-339. doi: 10.4236/ajor.2014.46032.
Meng, Z. , Shen, R. and Jiang, M. (2014) A Penalty Function Algorithm with Objective Parameters and Constraint Penalty Parameter for Multi-Objective Programming. American Journal of Operations Research, 4, 331-339. doi: 10.4236/ajor.2014.46032.
References
[1] Sawaragi, Y., Nakayama, H. and Tanino, T. (1985) Theory of Multiobjective Optimization. Academic Press, London. [2] White. D.J. (1984) Multiobjective Programming and Penalty Functions. Journal of Optimization Theory and Applications, 43, 583-599.
http://dx.doi.org/10.1007/BF00935007 [3] Sunaga, T., Mazeed, M.A. and Kondo, E. (1988) A Penalty Function Formulation for Interactive Multiobjective Programming Problems. Lecture Notes in Control and Information Sciences, 113, 221-230.
http://dx.doi.org/10.1007/BFb0042790 [4] Ruan, G.Z. and Huang, X.X. (1992) Weak Calmness and Weak Stability of Multiobjective Programming and Exact Penalty Functions. Journal of Mathematics and System Science, 12, 148-157. [5] Liu. J.C. (1996) -Pareto Optimality for Nondifferentiable Multiobjective Programming via Penalty Function. Journal of Mathematical Analysis and Applications, 198, 248-261.
http://dx.doi.org/10.1006/jmaa.1996.0080 [6] Huang, X.X. and Yang, X.Q. (2002) Nonlinear Lagrangian for Multiobjective Optimization to Duality and Exact Penalization. SIAM Journal on Optimization, 13, 675-692.
http://dx.doi.org/10.1137/S1052623401384850 [7] Chang, C.-T. and Lin, T.-C. (2009) Interval Goal Programming for S-Shaped Penalty Function. European Journal of Operational Research, 199, 9-20.
http://dx.doi.org/10.1016/j.ejor.2008.10.009 [8] Antczak, T. (2012) The Vector Exact l1 Penalty Method for Nondifferentiable Convex Multiobjective Programming Problems. Applied Mathematics and Computation, 218, 9095-9106.
http://dx.doi.org/10.1016/j.amc.2012.02.056 [9] Huang, X.X., Teo, K.L. and Yang, X.Q. (2006) Calmness and Exact Penalization in Vector Optimization with Cone Constraints. Computational Optimization and Applications, 35, 47-67.
http://dx.doi.org/10.1007/s10589-006-6441-5 [10] Huang, X.X. (2012) Calmness and Exact Penalization in Constrained Scalar Set-Valued Optimization. Journal of Optimization Theory and Applications, 154, 108-119.
http://dx.doi.org/10.1007/s10957-012-9998-4 [11] Meng, Z.Q., Shen, R. and Jiang, M. (2011) An Objective Penalty Functions Algorithm for Multiobjective Optimization Problem. American Journal of Operations Research, 1, 229-235.
http://dx.doi.org/10.4236/ajor.2011.14026 [12] Luque, M., Ruiz, F. and Steuer, R.E. (2010) Modi-Fied Interactive Chebyshev Algorithm (MICA) for Convex Multiobjective Programming. European Journal of Op-Erational Research, 204, 557-564.
http://dx.doi.org/10.1016/j.ejor.2009.11.011
[1] Sawaragi, Y., Nakayama, H. and Tanino, T. (1985) Theory of Multiobjective Optimization. Academic Press, London. [2] White. D.J. (1984) Multiobjective Programming and Penalty Functions. Journal of Optimization Theory and Applications, 43, 583-599.
http://dx.doi.org/10.1007/BF00935007 [3] Sunaga, T., Mazeed, M.A. and Kondo, E. (1988) A Penalty Function Formulation for Interactive Multiobjective Programming Problems. Lecture Notes in Control and Information Sciences, 113, 221-230.
http://dx.doi.org/10.1007/BFb0042790 [4] Ruan, G.Z. and Huang, X.X. (1992) Weak Calmness and Weak Stability of Multiobjective Programming and Exact Penalty Functions. Journal of Mathematics and System Science, 12, 148-157. [5] Liu. J.C. (1996) -Pareto Optimality for Nondifferentiable Multiobjective Programming via Penalty Function. Journal of Mathematical Analysis and Applications, 198, 248-261.
http://dx.doi.org/10.1006/jmaa.1996.0080 [6] Huang, X.X. and Yang, X.Q. (2002) Nonlinear Lagrangian for Multiobjective Optimization to Duality and Exact Penalization. SIAM Journal on Optimization, 13, 675-692.
http://dx.doi.org/10.1137/S1052623401384850 [7] Chang, C.-T. and Lin, T.-C. (2009) Interval Goal Programming for S-Shaped Penalty Function. European Journal of Operational Research, 199, 9-20.
http://dx.doi.org/10.1016/j.ejor.2008.10.009 [8] Antczak, T. (2012) The Vector Exact l1 Penalty Method for Nondifferentiable Convex Multiobjective Programming Problems. Applied Mathematics and Computation, 218, 9095-9106.
http://dx.doi.org/10.1016/j.amc.2012.02.056 [9] Huang, X.X., Teo, K.L. and Yang, X.Q. (2006) Calmness and Exact Penalization in Vector Optimization with Cone Constraints. Computational Optimization and Applications, 35, 47-67.
http://dx.doi.org/10.1007/s10589-006-6441-5 [10] Huang, X.X. (2012) Calmness and Exact Penalization in Constrained Scalar Set-Valued Optimization. Journal of Optimization Theory and Applications, 154, 108-119.
http://dx.doi.org/10.1007/s10957-012-9998-4 [11] Meng, Z.Q., Shen, R. and Jiang, M. (2011) An Objective Penalty Functions Algorithm for Multiobjective Optimization Problem. American Journal of Operations Research, 1, 229-235.
http://dx.doi.org/10.4236/ajor.2011.14026 [12] Luque, M., Ruiz, F. and Steuer, R.E. (2010) Modi-Fied Interactive Chebyshev Algorithm (MICA) for Convex Multiobjective Programming. European Journal of Op-Erational Research, 204, 557-564.
http://dx.doi.org/10.1016/j.ejor.2009.11.011