Abstract
In the absence of a clear “objective” value function, it is still possible in many cases to construct a domination cone according to which efficient (nondominated) solutions can be found. The relations between value functions and domination cones and between efficiency and optimality are analyzed here. We show that such cones must be convex, strictly supported and, frequently, closed as well. Furthermore, in most applications “potential” optimal solutions are equivalent to properly efficient points. These solutions can often be produced by maximizing with respect to a class of concave functions or, under convexity conditions, a class of affine functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K. Arrow, E. Barankin and D. Blackwell, “Admissible points of convex sets,” in: H.W. Kuhn and A.W. Tucker, eds.,Contribution to the Theory of Games (Princeton University Press, Princeton, NJ, 1953) pp. 87–91.
A. Ben-Israel, A. Ben-Tal and S. Zlobec,Optimality in Nonlinear Programming, a Feasible Directions Approach (Wiley, New York, 1981).
H. Benson, “An improved definition of proper efficiency for vector maximization with respect to cones,”Journal of Mathematical Analysis and Applications 71 (1979) 232–241.
G. Bitran and T. Magnanti, “The structure of admissible points with respect to cone dominance,”Journal of Optimization Theory and Applications 29 (1979) 573–614.
J. Borwein, “Proper efficient points for maximization with respect to cones,”SIAM Journal on Control and Optimization 15 (1977) 57–63.
Y. Censor, “Pareto optimality in multiobjective problems,”Applied Mathematics and Optimization 4 (1977) 41–59.
P.C. Fishburn,Utility Theory for Decision Making (Wiley, New York, 1970).
D. Gale,The Theory of Linear Economic Models (McGraw-Hill, New York, 1960).
W.B. Gearhart, “Compromise solutions and estimation of the noninferior set,”Journal of Optimization Theory and Applications 28 (1979) 29–47.
W.B. Gearhart, “Characterization of properly efficient solutions by generalized scalarization methods,”Journal of Optimization Theory and Applications 41 (1983) 491–502.
A. Geoffrion, “Proper efficiency and the theory of vector maximization,”Journal of Mathematical Analysis and Applications 22 (1968) 618–630.
I.V. Girsanov, “Lectures on mathematical theory of extremum problems,” in:Lecture Notes in Economics and Mathematical Systems (Springer, Berlin, 1972).
R. Hartley, “On cone-efficiency, cone-convexity, and cone-compactness,”SIAM Journal on Applied Mathematics 34 (1978) 211–222.
M. Henig, “Proper efficiency with respect to cones,”Journal of Optimization Theory and Applications 36 (1982) 387–407.
M. Henig, “A cone separation theorem,”Journal of Optimization Theory and Applications 36 (1982) 451–455.
H. Isermann, “Proper efficiency and the linear vector maximum problem,”Operations Research 22 (1974) 189–191.
J. Jahn, “A characterization of properly minimal elements of a set,”SIAM Journal on Control and Optimization 23 (1985) 649–656.
S. Karlin, “Mathematical methods and theory in games,” in:Programming and Economics, Vol. 1 (Addison-Wesley, Reading, MA, 1959).
D. Krantz, D. Luce, D. Suppes and D. Tversky,Foundation of Measurements, Vol. 1 (Academic Press, New York, 1971).
H.W. Kuhn and A.W. Tucker, “Nonlinear programming,” in: J. Neyman, ed.,Second Berkeley Symposium on Mathematical Statistics and Probability (University of California Press, Berkeley, CA, 1950) pp. 481–492.
R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).
R. Soland, “Multicriteria optimization: A general characterization of efficient solution,”Decision Sciences 10 (1970) 26–38.
D.J. White, “Optimaltiy and efficiency I,”European Journal of Operational Research 4 (1980) 346–356.
P.L. Yu, “Cone convexity, cone extreme points, and nondominated solutions in decision problems with multi objectives,”Journal of Optimization Theory and Applications 14 (1974) 319–377.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Henig, M.I. Value functions, domination cones and proper efficiency in multicriteria optimization. Mathematical Programming 46, 205–217 (1990). https://doi.org/10.1007/BF01585738
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01585738