Hostname: page-component-77c89778f8-gvh9x Total loading time: 0 Render date: 2024-07-21T09:07:55.733Z Has data issue: false hasContentIssue false
  • English
  • Français

Proper efficiency and duality for vector valued optimization problems

Published online by Cambridge University Press:  09 April 2009

T. Weir  and
B. Mond
T. Weir
Affiliation:
Department of Mathematics, University College, Australian Defence Force Academy, Campbell A. C. T. 2600, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The duality results of Wolfe for scalar convex programming problems and some of the more recent duality results for scalar nonconvex programming problems are extended to vector valued programs. Weak duality is established using a ‘Pareto’ type relation between the primal and dual objective functions.

Keywords

vector valued programmingefficiencyproper efficiencyduality

MSC classification

Secondary: 90C31: Sensitivity, stability, parametric optimization 90C30: Nonlinear programming
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 43 , Issue 1 , August 1987 , pp. 21 - 34
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Bector, C. R. and Klassen, J. E., ‘Duality for a nonlinear programming problem’, Utilitas Math. 11 (1977), 87–99.Google Scholar
[2]Bitran, G., ‘Duality in nonlinear multiple criteria optimization problems’, J. Optim. Theory Appl. 35 (1981) 367–406.CrossRefGoogle Scholar
[3]Corley, B. D., ‘Duality theory for maximations with respect to cones’, J. Math. Anal. Appl. 84 (1981), 560–568.CrossRefGoogle Scholar
[4]Craven, B. D. ‘Strong vector minimization and duality’, Z. Angew. Math. Mech. 60 (1980), 1–5.Google Scholar
[5]Crouzeix, J. P., ‘A duality framework in quasiconvex programming’, Generalized concavity in optimization and economics edited by Schaible, S. and Ziemba, W. T., pp. 207–226 (Academic Press, New York, 1981).Google Scholar
[6]Fenchel, W., ‘Convex cones, sets and functions’ (Lecture notes, Armed Services Technical Information Agency, AD Number 22695, Princeton University, 1953).Google Scholar
[7]Geoffrion, A. M., ‘Proper efficiency and the theory of vector maximization’, J. Math. Anal. Appl. 22, (1968), 618–630.Google Scholar
[8]Geoffrion, A. M., ‘Duality in nonlinear programming: a simplified applications oriented development], SIAM Rev. 13 (1971), 1–37.Google Scholar
[9]Ivanov, e. H. and Nehse, R., ‘Some results on dual vector optimization problems’, Optimization 16 (1985), 505–517.CrossRefGoogle Scholar
[10]Luenberger, D. G., ‘Quasiconvex programming’, SIAM J. Appl. Math. 16 (1968), 1090–1095.Google Scholar
[11]Mahajan, D. G. and Vartak, M. N., ‘Generalization of some duality theorems in nonlinear programming’, Math. Programming 12 (1977), 293–317.Google Scholar
[12]Mangasarian, O. L., Nonlinear programming (McGraw-Hill, New York, 1969).Google Scholar
[13]Mangasarian, O. L. and Ponstein, J., ‘Minimax and duality in nonlinear programming’, J. Math. Anal. Appl. 11 (1965), 504–518.CrossRefGoogle Scholar
[14]Mond, B. and Weir, T., ‘Generalized concavity and duality’, Generalized concavity in optimization and economics edited by Schaible, S. and Ziemba, W. T., pp. 263–279 (Academic Press, New York, 1981).Google Scholar
[15]Pareto, V., Cours d'economie politique (Rouge, Lausanne, 1896).Google Scholar
[16]Rockafellar, R. T., Conjugate duality and optimization (Regional conference series in applied mathematics, No. 16, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1974).CrossRefGoogle Scholar
[17]Stoer, J., ‘Duality in nonlinear programming and the minimax theorem’, Numer. Math. 5 (1963), 371–379.Google Scholar
[18]Tanino, T. and Sawaragi, Y., ‘Duality theory in multiobjective programming’, J. Optim. Theory Appl. 27 (1979), 509–529.CrossRefGoogle Scholar
[19]Tanino, T. and Sawaragi, Y., ‘Conjugate maps and duality in multi-objective optimization’, J. Optim. Theory Appl. 31 (1980), 473–499.CrossRefGoogle Scholar
[20]Whinston, A., ‘Conjugate functions and dual programs, Naval Res. Logist. Quart. 12 (1965), 315–322.Google Scholar
[21]White, D. G., ‘Concepts of proper efficiency’, European J. Oper. Res. 13 (1983), 180–188.CrossRefGoogle Scholar
[22]White, D. G., ‘Vector maximization and Lagrange multipliers’, Math. Programming 31 (1985), 192–205.CrossRefGoogle Scholar
[23]Wolfe, P., ‘A duality theorem for nonlinear programming’, Quart. Appl. Math. 19 (1961), 239–244.CrossRefGoogle Scholar