zbMATH — the first resource for mathematics

Multiobjective bilevel optimization. (English) Zbl 1198.90347
This paper deals with nonlinear non-convex multi-objective bi-level optimization problems which are discussed using an optimistic approach. The author aims to obtain a good approximation of the feasible set of the upper level function by expressing it as the set of minimal solutions of a multi-objective optimization problem. To solve this problem he applies the scalarization approach of A. Pascoletti and P. Serafini [J. Optimization Theory Appl. 42, 499–524 (1984; Zbl 0505.90072)].
For generating the approximation mentioned above, the author uses sensitivity results for controlling the parameters of the corresponding scalarization problem adaptively. This sensitivity results are used again for solving the upper level problem in an iterative process. Thus, not only one minimal solution but an approximation of the whole efficient set of the multi-objective bilevel optimization problem is determined.
The proposed numerical method (without convexity assumptions) demands twice continuously differentiable functions and appropriate solvers for determining global solutions of the scalar problems.
Finally, an academic example and a topological problem arising in an application are solved with an algorithm designed for the case of a bi-criteria lower and upper level problem and a one-dimensional upper level variable.

90C29 Multi-objective and goal programming
90C31 Sensitivity, stability, parametric optimization
90C59 Approximation methods and heuristics in mathematical programming
PDF BibTeX Cite
Full Text: DOI
[1] Abo-Sinna M.: A bi-level non-linear multi-objective decision making under fuzziness. Opsearch 38(5), 484–495 (2001) · Zbl 1278.90493
[2] Alt W.: Parametric optimization with applications to optimal control and sequential quadratic programming. Bayreuther Math. Schr. 35, 1–37 (1991) · Zbl 0734.90094
[3] Bard, J.F.: Practical Bilevel Optimization. Algorithms and Applications. Nonconvex Optimization and Its Applications, vol. 30. Kluwer, Dordrecht (1998) · Zbl 0943.90078
[4] Bonnel H., Morgan J.: Semivectorial bilevel optimization problem: Penalty approach. J. Optim. Theory Appl. 131(3), 365–382 (2006) · Zbl 1205.90258
[5] Das I., Dennis J.: A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Struct. Optim. 14, 63–69 (1997)
[6] Das I., Dennis J.: Normal-boundary intersection: A new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM J. Optim. 8(3), 631–657 (1998) · Zbl 0911.90287
[7] Dempe S.: Foundations of Bilevel Programming. Nonconvex Optimization and Its Applications, vol. 61. Kluwer, Dordrecht (2002) · Zbl 1038.90097
[8] Dempe S.: Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52(3), 333–359 (2003) · Zbl 1140.90493
[9] Dempe, S.: Bilevel programming–a survey. Preprint 2003-11, Fakultät für Mathematik und Informatik, TU Bergakademie Freiberg, Germany (2003) · Zbl 1140.90493
[10] Dinkelbach, W., Dürr, W.: Effizienzaussagen bei Ersatzprogrammen zum Vektormaximumproblem, iv. Oberwolfach-Tag. Operations Res 1971. Oper. Res. Verf. 12:69–77 (1972) · Zbl 0253.90056
[11] Ehrgott, M.: Multicriteria optimisation. Lect. Notes Econ. math. Syst, vol. 491. Springer, Berlin (2000) · Zbl 0956.90039
[12] Eichfelder, G.: Parametergesteuerte Lösung nichtlinearer multikriterieller Optimierungsprobleme. PhD Thesis, Univ. Erlangen-Nürnberg, Germany (2006)
[13] Eichfelder, G.: Scalarizations for adaptively solving multi-objective optimization problems. Comput. Optim. Appl. (2007) doi: 10.1007/s10589-007-9155-4 · Zbl 1184.90152
[14] Eichfelder, G.: An adaptive scalarization method in multi-objective optimization. SIAM J. Optim. (to appear) · Zbl 1187.90252
[15] Fiacco, A.V.: Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Mathematics in Science and Engineering, vol. 165. Academic Press, London (1983) · Zbl 0543.90075
[16] Fliege J.: Gap-free computation of Pareto-points by quadratic scalarizations. Math. Methods Oper. Res. 59(1), 69–89 (2004) · Zbl 1131.90054
[17] Fliege J., Vicente LN: Multicriteria approach to bilevel optimization. J. Optim. Theory Appl. 131(2), 209–225 (2006) · Zbl 1139.90417
[18] Haimes Y., Lasdon L., Wismer D.: On a bicriterion formulation of the problems of integrated system identification and system optimization. IEEE Trans. Syst. Man Cybern. 1, 296–297 (1971) · Zbl 0224.93016
[19] Hillermeier C., Jahn J.: Multiobjective optimization: survey of methods and industrial applications. Surv. Math. Ind. 11, 1–42 (2005) · Zbl 1160.90651
[20] Jahn J.: Vector Optimization: Theory, Applications and Extensions. Springer, Berlin (2004) · Zbl 1055.90065
[21] Jahn J.: Multiobjective search algorithm with subdivision technique. Comput. Optim. Appl. 35, 161–175 (2006) · Zbl 1153.90533
[22] Jahn J., Merkel A.: Reference point approximation method for the solution of bicriterial nonlinear optimization problems. J. Optim. Theory Appl. 74(1), 87–103 (1992) · Zbl 0795.90056
[23] Jittorntrum K.: Solution point differentiability without strict complementarity in nonlinear programming. Math. Program. Study 21, 127–138 (1984) · Zbl 0571.90080
[24] Kim I., de Weck O.: Adaptive weighted sum method for bi-objective optimization. Struct. Multidisciplinary Optim. 29, 149–158 (2005)
[25] Lin J.G.: On min-norm and min-max methods of multi-objective optimization. Math. Program. 103(1), 1–33 (2005) · Zbl 1079.90123
[26] Loridan P.: {\(\epsilon\)}-solutions in vector minimization problems. J. Optim. Theory Appl. 43, 265–276 (1984) · Zbl 0531.90091
[27] Marglin S.: Public Investment Criteria. MIT Press, Cambridge (1967)
[28] Miettinen K.M.: Nonlinear Multiobjective Optimization. Kluwer, Boston (1999) · Zbl 0949.90082
[29] Nishizaki I., Sakawa M.: Stackelberg solutions to multiobjective two-level linear programming problems. J. Optim. Theory Appl. 103(1), 161–182 (1999) · Zbl 0945.90057
[30] Osman M., Abo-Sinna M., Amer A., Emam O.: A multi-level nonlinear multi-objective decision-making under fuzziness. Appl. Math. Comput. 153(1), 239–252 (2004) · Zbl 1049.90032
[31] Pascoletti A., Serafini P.: Scalarizing vector optimization problems. J. Optim. Theory Appl. 42(4), 499–524 (1984) · Zbl 0505.90072
[32] Prohaska, J.: Optimierung von Spulenkonfigurationen zur Bewegung magnetischer Sonden. Diplomarbeit, Univ. Erlangen-Nürnberg, Germany (2005)
[33] Ruzika S., Wiecek M.: Approximation methods in multiobjective programming. J. Optim. Theory Appl. 126(3), 473–501 (2005) · Zbl 1093.90057
[34] Sawaragi Y., Nakayama H., Tanino T.: Theory of Multiobjective Optimization. Number 176 in Mathematics in science and engineering. Academic Press, London (1985) · Zbl 0566.90053
[35] Schandl B., Klamroth K., Wiecek M.M.: Norm-based approximation in bicriteria programming. Comput. Optim. Appl. 20(1), 23–42 (2001) · Zbl 0983.90081
[36] Shi X., Xia H.: Interactive bilevel multi-objective decision making. J. Oper. Res. Soc. 48(9), 943–949 (1997) · Zbl 0892.90200
[37] Shi X., Xia H.: Model and interactive algorithm of bi-level multi-objective decision-making with multiple interconnected decision makers. J. Multi-Criteria Decis. Anal. 10, 27–34 (2001) · Zbl 0985.90054
[38] Staib T.: On two generalizations of Pareto minimality. J. Optim. Theory Appl. 59(2), 289–306 (1988) · Zbl 0628.90076
[39] Teng C.-X., Li L., Li H.-B.: A class of genetic algorithms on bilevel multi-objective decision making problem. J. Syst. Sci. Syst. Eng. 9(3), 290–296 (2000)
[40] Tuy H., Migdalas A., Hoai-Phuong N.: A novel approach to bilevel nonlinear programming. J. Global Optim. 38(4), 527–554 (2007) · Zbl 1145.90083
[41] Vicente L.N., Calamai P.H.: Bilevel and multilevel programming: A bibliography review. J. Glob. Optim. 5(3), 291–306 (1994) · Zbl 0822.90127
[42] Yin Y.: Multiobjective bilevel optimization for transportation planning and management problems. J. Adv. Transp. 36(1), 93–105 (2000)
[43] Younes, Y.: Studies on discrete vector optimization. Dissertation, University of Demiatta, Egypt (1993)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.