Benson's algorithm for nonconvex multiobjective problems via nonsmooth Wolfe duality

Document Type : Research Paper

Author

Department of Mathematics, University of Isfahan, Isfahan, Iran.

Abstract

‎In this paper‎, ‎we propose an algorithm to obtain an approximation set of the (weakly) nondominated points of nonsmooth multiobjective optimization problems with equality and inequality constraints‎. ‎We use an extension of the Wolfe duality to construct the separating hyperplane in Benson's outer algorithm for multiobjective programming problems with subdifferentiable functions‎. ‎We also formulate an infinitive approximation set of the (weakly) nondominated points of biobjective optimization problems‎. ‎Moreover‎, ‎we provide some numerical examples to illustrate the advantage of our algorithm.

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Q.H. Ansari and J.C. Yao (eds.) Recent Developments in Vector Optimization, Springer, Berlin, 2012.
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.
F.H. Clarke, Y.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth Analysis and Control Theory, Grad. Texts in Math. 178, Springer-Verlag, New York, 1998.
M. Ehrgott, A. Lohne and L. Shao, A dual variant of Benson's outer approximation algorithm, J. Global Optim. 52 (2012), no. 4, 757--778.
M. Ehrgott, L. Shao and A. Schobel, An approximation algorithm for convex multiobjective programming problems, J. Global Optim. 50 (2011), no. 3, 397--416.
G. Eichfelder, Scalarizations for adaptively solving multi-objective optimization problems, Comput. Optim. Appl. 44 (2009), no. 2, 249--273.
H. Eschenauer, J. Koski and A. Osyczka, Multicriteria Design Optimization Procedures and Applications, Springer-Verlag, Berlin, 1990.
J. Fernandez and B. Toth, Obtaining the efficient set of nonlinear biobjective optimization problems via interval branch-and-bound methods, Comput. Optim. Appl. 42 (2009), no. 3, 393--419.
P. Lindroth, M. Patriksson and A.B. Stromberg, Approximating the Pareto optimal set using a reduced set of objective functions, European J. Oper. Res. 207 (2010), no. 3, 1519--1534.
G.P. Liu, J.B. Yang and J.F. Whidborne, Multiobjective Optimization and Control, Research Studies Press, Bristol, 2002.
A. Lohne, B. Rudloff and F. Ulus, Primal and dual approximation algorithms for convex vector optimization problems, J. Global Optim. 60 (2014), no. 4, 713--736.
D. T. Luc, Theory of Vector Optimization, Lecture Notes in Econom. and Math. Systems 319, Springer-Verlag, Berlin, 1989.
O.L. Mangasarian, Nonlinear Programming, McGraw-Hill, New York, 1969.
K.M. Miettinen, Nonlinear Multiobjective Optimizatoin, Kluwer Academic Publ. Boston, 1999.
S. Mititelu, A survey on optimality and duality in nonsmooth programming, in: Generalized Convexity, pp. 211--225, Lecture Notes in Econom. and Math. Systems 405, Springer, Berlin, 1994.
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.
R.B. Statnikov and J.B. Matusov, Multicriteria Optimization and Engineering, Chapman & Hall, New York, 1995.
S. Ruzika and M.M. Wiecek, Approximation methods in multiobjective programming, J. Optim. Theory Appl. 126 (2005), no. 3, 473--501.