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

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 x⁰ 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.