Scalarization and Stability

Abstract

The main difference between scalar optimization and vector optimization lies in the underlying preference orders on the space concerned. In the scalar case, as the functions to be maximized or minimized are valued in the one dimensional space where a complete order is given, it can be decided, for each pair of alternatives, which of them is prefered. However, this important feature is no longer valid in the vector case, because the preference orders, as we have seen, are generally not complete. To overcome difficulties caused by the noncompleteness of the orders, techniques which convert vector problems into appropriate scalar problems axe widely applied. In other words, given a vector optimization problem

$$ \begin{array}{*{20}{c}} {\min F\left( x \right)} \\ {s.t.{\kern 1pt} {\kern 1pt} x \in X} \\ \end{array} $$

, where as before F is a set-valued map from a nonempty set X to a vector space E ordered by a convex cone C , one tries to find another problem, say

$$ \begin{array}{*{20}{c}} {\min G\left( x \right)} \\ {s.t.{\kern 1pt} {\kern 1pt} x \in X} \\ \end{array} $$

, where G is a set-valued map from X to R, so that the latter problem is much likely easier to be dealt and provides optimal solutions of the former problem. This chapter is devoted to the method mentioned above. In Section 1 we develop separation technique for nonconvex sets by means of monotonic functions. Section 2 deals with scalar representations. Several representations are provided which preserve the linearity, convexity and quasiconvexity properties of the original problems.