Abstract
The Lagrangean function for scalar constrained optimisation problems is extended in a directly analogous manner to constrained vector optimisation problems. Some simple saddle point results are presented for vector maxima sets. Conditions are given for the characterisation of the vector maximum set of the original vector problem in terms of the vector maximum sets with respect to the vector Lagrangeans.
Finally some attention is given to Lagrangean relaxation for vector optimisation problems as an extension of a result of Everett.
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References
G. Bitran, “Duality in nonlinear multiple criteria optimisation problems”,Journal of Optimisation Theory and Applications 35 (1981) 367–406.
S. Brumelle, “Duality for multiple objective convex programs”,Mathematics of Operations Research 6 (1981) 159–172.
A. Charnes and W.W. Cooper,Management models and industrial applications of linear programming I (Wiley, 1981).
H. Everett, “Generalised Lagrange multiplier method for solving problems of optimal allocation of resources”,Operation Research 11 (1963).
A.M. Geoffrion,Perspectives in optimisation (Addison-Wesley, Reading, MA, 1972).
A.M. Geoffrion, “Lagrangean relaxation for integer programming”,Mathematical Programming Study 2 (1974) 82–114.
S. Karlin,Mathematical methods and theory in games, programming and economics (Addison-Wesley, Reading, MA, 1959).
H. Kawasaki, “A duality theorem in multi-objective nonlinear programming”,Mathematics of Operations Research 7 (1982) 95–110.
G.L. Nemhauser and Z. Ullmann, “A note on the generalised Lagrange multiplier solution to an integer programming problem”,Operations Research 16 (1968) 450–453.
V. Pareto,Cours d’economie politique (Rouge, 1896).
W. Rödder, “A generalised saddle point theory”,European Journal of Operational Research 1 (1977) 55–59.
Y. Sarawagi and T. Tanino, “Duality theory in multiobjective programming”,Journal of Optimisation Theory and Applications 27 (1979) 509–529.
Y. Sarawagi and T. Tanino, “Conjugate maps and duality in multiobjective optimisation”,Journal of Optimisation Theory and Applications 31 (1980) 473–499.
W. Stadler, “A survey of multicriteria optimisation in the vector maximum problem, part I: 1776–1960”,Journal of Optimisation Theory and Applications 29 (1979) 1–52.
D.J. White,Optimality and efficiency (Wiley, New York, 1982).
D.J. White, “A note on the generalised Lagrange solution to an integer programming problem”, Note in Decision Theory 16, Department of Decision Theory, Machester University (Manchester, England 1974).
D.J. White, “A branch and bound method for multiobjective Boolean problems”,European Journal of Operational Research 15 (1984), 126–130.
D.J. White, “Multi-objective programming and penalty functions”,Journal of Optimisation Theory 43 (1984) 583–600.
V.I. Zangwill,Non-linear programming (Prentice-Hall, Englewood Cliffs, NJ, 1969).
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White, D.J. Vector maximisation and lagrange multipliers. Mathematical Programming 31, 192–205 (1985). https://doi.org/10.1007/BF02591748
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DOI: https://doi.org/10.1007/BF02591748