Restricting the parameter set of the Pascoletti-Serafini scalarization

Document Type: ORO2013

Author

Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, No. 424, Hafez Ave, 15914, Tehran, Iran

Abstract

‎A common approach to determine efficient solutions of a multiple objective optimization problem‎ ‎is reformulating it to a parameter dependent scalar optimization problem‎. ‎This reformulation is called scalarization approach‎. Here, a well-known scalarization approach named Pascoletti-Serafini scalarization is considered‎. First, some difficulties of this scalarization are discussed and then removed by restricting the parameter set‎. A method is presented to convert a space ordered by a specific‎ ‎ordering cone to an equivalent space ordered by the natural ordering cone‎. ‎Utilizing the presented conversion‎, ‎all confirmed results and theorems for‎ ‎multiple objective optimization problems ordered by the natural‎ ‎ordering cone can be extended to multiple objective optimization‎ ‎problems ordered by specific ordering cones.

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C. D. Aliprantis, P. K. Monteiro and R. Tourky, Non-marketed options, non-existence of equilibria, and non-linear prices, J. Econom. Theory 114 (2004), no. 2, 345--357.

J. M. Borwein, The geometry of Pareto efficiency over cones, Math. Operationsforsch. Statist. Ser. Optim. 11 (1980), no. 2, 235--248.

V. Chankong and Y. Haimes, Multiobjective Decision Making: Theory and Methodology, Elsevier Science Publishing Co. New York, 1983.

I. Das and J. E. Dennis, Normal-boundary intersection: A new method for generating the Pareto surface in nonlinear multicriteria optimization problems, SIAM J. Optim. 8 (1998), no. 3, 631--657.

G. Dorini, F. Di Pierro, D. Savic and A. B. Piunovskiy, Neighbourhood search for constructing pareto sets, Math. Methods Oper. Res. 65 (2007), no. 2, 315--337.

M. Ehrgott, Multicriteria Optimization, Second edition. Springer-Verlag, Berlin, 2005.

M. Ehrgott and M. Wiecek, Multiobjective Programming, Multiple Criteria Decision Analysis, State of the Art Surveys, Springer, New York, 2005.

M. Ehrgott and S. Ruzika, Improved ϵ-constraint method for multiobjective programming, J. Optim. Theory Appl. 138 (2008), no. 3, 375--396.

G. Eichfelder, Adaptive Scalarization Methods in Multiobjective Optimization, Springer-Verlag, Berlin, 2008.

C. Gerstewitz and P. Weidner, Nonconvex separation theorems and some applications in vector optimization, J. Optim. Theory Appl. 67 (1990), no. 2, 297--320.

C. Gerstewitz and E. Iwanow, Duality fur nichtkonvexe vektoroptimierungsprobleme, (German) Wiss. Z. Tech. Hochsch. Ilmenau 31 (1985), no. 2, 61--81.

D. Gourion and D. Luc, Finding efficient solutions by free disposal outer approximation, SIAM J. Optim. 20 (2010), no. 6, 2939--2958.

Y. Haimes, L. Lasdon and D. Wismer, On a bicriterion formulation of the problems of integrated system identi_cation and system optimization, IEEE Trans. Syst. Man. Cybern. Syst. (1971) 296--297.

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

I. Kaliszewski, A modified weighted Tchebycheff metric for multiple objective programming, Comput. Oper. Res. 14 (1987), no. 4, 315--323.

I. Kaliszewski, A theorem on nonconvex functions and its application to vector optimization, European J. Oper. Res. 80 (1995) 439--449.

E. Khorram, K. Khaledian and M. Khaledyan, A numerical method for constructing the Pareto front of multi-objective optimization problems, J. Comput. Appl. Math. 261 (2014) 158--171.

D. Li, Convexification of a noninferior frontier, J. Optim. Theory Appl. 88 (1996), no. 1, 177--196.

D. T. Luc, Theory of Vector Optimization, Lecture Notes in Econom. and Math. Systems, 319, Springer-Verlag, Berlin, 1989.

A. Messac and C. A. Mattson, Normal constraint method with guarantee of even representation of complete Pareto frontier, AIAA J. 42 (2004) 2101--2111.

K. M. Miettinen, Nonlinear Multiobjective Optimization, Kluwer Academic Publishers, Boston, 1999.

A. Pascoletti and P. Sera_ni, Scalarizing vector optimization problems, J. Optim. Theory Appl. 42 (1984), no. 4, 499--524.

R. Ramesh, M. H. Karwan and S. Zionts, Theory of convex cones in multicriteria decision making, Multi-attribute decision making via O.R.-based expert systems, Ann. Oper. Res. 16 (1988), no. 1-4, 131--147.

A. M. Rubinov, Sublinear operator and theirs applications, Russian Math. Surveys 32 (1977), no. 4, 113--175.

Y. Sawaragi, H. Nakayama and T. Tanino, Theory of Multiobjective Optimization, Academic Press, Orlando, 1985.

C. Tammer and C. Zalinescu, Lipschitz properties of the scalarization function and applications, Optimization 59 (2010), no. 2, 305--319.