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Letter to the Editor—Improper Solutions of the Vector Maximum Problem

Author

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  • Allen Klinger

    (The Rand Corporation, Santa Monica, California)

Abstract

Many problems in operations research, engineering, and economics require that several objectives be maximized simultaneously. It is well known that ‘maximizing’ the vector whose components are these objectives can yield more solutions than maximizing one linear combination of the objectives. Kuhn and Tucker (Kuhn, H. W., A. W. Tucker. 1951. Nonlinear programming. Proc. Second Berkeley Symp. on Math. Stat. and Prob. University of California Press, Berkeley, Calif. 1951.) discuss the vector maximum problem and derive necessary and sufficient conditions for a vector x 0 to be a proper solution. However, their basic paper presents only a definition of ‘proper’ and a particular vector maximum problem for which an ‘improper’ solution has an undesirable property. The purpose of this note is to show that all ‘improper’ solutions have this undesirable property, thus justifying the calculation of only the ‘proper’ solutions.

Suggested Citation

  • Allen Klinger, 1967. "Letter to the Editor—Improper Solutions of the Vector Maximum Problem," Operations Research, INFORMS, vol. 15(3), pages 570-572, June.
    • Handle: RePEc:inm:oropre:v:15:y:1967:i:3:p:570-572
    DOI: 10.1287/opre.15.3.570
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    Cited by:

    1. Adel A. Aly & Boubekeur Rahali, 1990. "Analysis of a bicriteria location model," Naval Research Logistics (NRL), John Wiley & Sons, vol. 37(6), pages 937-944, December.
    2. Oliver Stein & Maximilian Volk, 2023. "Generalized Polarity and Weakest Constraint Qualifications in Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 198(3), pages 1156-1190, September.
    3. Savin Treanţă & Tadeusz Antczak & Tareq Saeed, 2023. "Connections between Non-Linear Optimization Problems and Associated Variational Inequalities," Mathematics, MDPI, vol. 11(6), pages 1-12, March.

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