An algorithm for approximating nondominated points of convex multiobjective optimization problems

Ghaznavi, M.; Azizi, Z.

An algorithm for approximating nondominated points of convex multiobjective optimization problems

Document Type : Research Paper

Authors

M. Ghaznavi ¹
Z. Azizi ²

¹ Faculty of Mathematics‎, ‎Shahrood University of Technology‎, ‎Shahrood‎, ‎Iran

² Faculty of Mathematics‎, ‎Shahrood University of Technology‎, ‎Shahrood‎, ‎Iran.

Abstract

‎In this paper‎, ‎we present an algorithm for generating approximate nondominated points of a multiobjective optimization problem (MOP)‎, ‎where the constraints and the objective functions are convex‎. ‎We provide outer and inner approximations of nondominated points and prove that inner approximations provide a set of approximate weakly nondominated points‎. ‎The proposed algorithm can be applied for differentiable or nondifferentiable convex MOPs‎. ‎To illustrate efficiency of the proposed algorithm for convex MOPs‎, ‎we provide numerical examples.

Keywords

Main Subjects

90-XX Operations Research, Mathematical Programming

References

Q.H. Ansari and J.C. Yao (eds.), Recent Developments in Vector Optimization, Springer, Berlin, 2011.

P. Armand, Finding all maximal efficient faces in multiobjective linear programming, Math. Program. 61 (1993), no. 3, 357--375.

M. Beldiman, E. Panaitescu and L. Dogaru, Approximate quasi efficient solutions in multiobjective optimization, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 51(99) (2008), no. 2, 109--121.

H. P. Benson, Hybrid approach for solving multiple-objective linear programs in outcome space, J. Optim. Theory Appl. 98 (1998), no. 1, 17-35.

H. P. Benson, An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem, J. Global Optim. 13 (1998), no. 1, 1-24.

A. Chinchuluun and P.M. Pardalos, A survey of recent developments in multiobjective optimization, Ann. Oper. Res. 154 (2007), no. 1, 29--50.

M. Ehrgott, Multicriteria Optimization, Springer, Berlin, 2005.

M. Ehrgott, L. Shao and A. Schobel, An approximation algorithm for convex multiobjective programming problems, J. Global Optim. 50 (2011), no. 3, 397--416.

A. Engau and M.M. Wiecek, Generating "-efficient solutions in multiobjective programming, Eur. J. Oper. Res. 177 (2007), no. 3, 1566--1579.

Y. Gao and X. Yang, Optimality conditions for approximate solutions in vector optimization problems, J. Ind. Manag. Optim. 7 (2011), no. 2, 483--496.

B.A. Ghaznavi-ghosoni and E. Khorram, On approximating weakly/properly efficient solutions in multiobjective programming, Math. Comput. Modelling 54 (2011), no. 11, 3172--3181.

B.A. Ghaznavi-ghosoni, E. Khorram and M. Soleimani-Damaneh, Scalarization for characterization of approximate strong/weak/proper efficiency in multiobjective optimization, Optimization 62 (2012), no. 6, 703--720.

M. Ghaznavi, Optimality conditions via scalarization for approximate quasi efficiency in multiobjective optimization, Filomat 31 (2017), no. 3, 671--680.

C. Gutierrez, B. Jimenez, V. Novo, On approximate solutions in vector optimization problems via scalarization, Comput. Optim. Appl. 35 (2006), no. 3, 305-324.

J. Jahn, Vector Optimization: Theory, Applications and Extensions, Springer, Berlin, 2004.

K. Khaledian, E. Khorram and B. Karimi, Characterizing "-properly efficient solutions, Optim. Methods Softw. 30 (2015), no. 3, 583--593.

N.T.B. Kim and D.T. Luc, Normal cones to a polyhedral convex set and generating efficient faces in linear multiobjective programming, Acta Math. Vietnam. 55 (2000), no 1, 101--124.

N.T.B. Kim, N.T. Thien and L.Q. Thuy, Generating all efficient extreme solutions in multiple objective linear programming problem and its application to multiplicative programming, East-West J. Math. 10 (2008), no. 1, 1--14.

D.S. Kim and T.Q. Son, "-optimal solutions in nonconvex semi-infinite programs with support functions, Fixed Point Theory Appl. 2011 (2011), Article ID 175327, 13 pages.

E. Kiyani and M. Soleimani-Damaneh, Approximate proper efficiency on real linear vector spaces, Pac. J. Optim. 10 (2014), no. 4, 715--734.

Z. Li and S. Wang, "􀀀approximation solutions in multiobjective optimization, Optimization 44 (1998) 161--174.

Z.H. Lin, D.L. Zhu and Z.P. Sheng, Finding a minimal efficient solution of a convex multiobjective program, J. Optim. Theory Appl. 118 (2003), no. 3, 587--600.

P. Loridan, "-solutions in vector minimization problems, J. Optim. Theory Appl. 43 (1984), no. 2, 265--276.

K. Miettinen, Nonlinear Multiobjective Optimization, Kluwer Academic, Dordrecht, 1999.

K. Miettinen, M.M. Makela, K. Kaario, Experiments with classification-based scalarizing functions in interactive multiobjective optimization. Eur. J. Oper. Res. 175 (2006) 931-947.

L. Pourkarimi, M.A. Yaghoobi and M. Mashinchi, Determining maximal efficient faces in multiobjective linear programming problem, J. Math. Anal. Appl. 354 (2009), no. 1, 234--248.

N. Rastegar and E. Khorram, A combined scalarizing method for multiobjective programming problems, European J. Oper. Res. 236 (2014), no. 1, 229--237.

L. Shao and M. Ehrgott, Approximately solving multiobjective linear programmes in objective space and an application in radiotherapy treatment planning, Math. Methods Oper. Res. 68 (2008), no. 2, 257--276.

M. Soleimani-Damaneh, On some multiobjective optimization problems arising in biology, Int. J. Comput. Math. 88 (2011), no. 6, 1103--1119.

W. Song and G.M. Yao, Homotopy method for a general multiobjective programming problem, J. Optim. Theory Appl. 138 (2008), no. 1, 139--153.

L.T. Thuy, N.T. Bach Kim and N.T. Thien, Generating efficient outcome points for convex multiobjective programming problems and its application to convex multiplicative programming, J. Appl. Math. 2011 (2011), Article ID 464832, 21 pages.

H. Tuy and N.D. Nghia, Reverse polyblock approximation for generalized multiplicative/fractional programming, Vietnam J. Math. 31 (2003), no. 4, 391--402.

R.X. Yue and Y. Gao, Scalarizations for approximate quasi efficient solutions in multiobjective optimization problems, J. Oper. Res. Soc. China 3 (2015) 69--80.

M. Zarepisheh, P.M. Pardalos, An equivalent transformation of multiobjective optimization problems, Ann. Oper. Res. 249 (2017), no. 1-2, 5--15.

K.Q. Zhao, G. Chen and X. Yang, Approximate proper efficiency in vector optimization, Optimization 64 (2015), no. 8, 1777--1793.

K.Q. Zhao, Y.M. Xia and X.M. Yang, Nonlinear scalarization characterizations of E-efficiency in vector optimization, Taiwanese J. Math. 19 (2015), no. 2, 455--466.

K.Q. Zhao and X.M. Yang, E-Benson proper efficiency in vector optimization, Optimization 64 (2015), no. 4, 739--752.

K.Q. Zhao, X.M. Yang and J.W. Peng, Weak E-optimal solution in vector optimization, Taiwanese J. Math. 17 (2013) 1287--1302.