Eigenvalues, Adjoints, and Conjugates of Set-Valued Sublinear Functions

Abstract

When one thinks about a mathematical function, one often conjures up a picture of its graph. For example, one usually identifies the function f(x) = x 2 with a graph of a parabola. In the field of mathematical analysis, the most basic function is the linear function. It is additive and homogenous. The graph of this function is basically a straight line in the plane. A related function to the linear function is an affine function, which is simply a linear function plus a constant. Based on the Taylor expansion, one can actually approximate any diffrentiable function by use of affine functions. When dealing with convex functions, functions whose epigraphs are convex sets, it is important to consider affine functions, given that each ’reasonable’ (i.e. proper and closed) convex function can be written as the supremum (basically, a maximum) of its affine minorants (the affine functions ’below’ the function). In essence, one can piece together many affine functions to construct a convex function. What is so important about convex functions? This is the first question one may ask when approaching convex analysis. Optimization theory is one major answer. Optimizing functions subject to constraints becomes incredibly easier when dealing with convex functions. This of course has many applications in areas such as economics and finance. As such, R.T. Rockafellar’s book ”Convex Analysis” published in 1970 is a seminal work in the field of optimization theory, for it lays down a whole theory for convex functions. In fact, Rockafellar, elsewhere [21], says ”the great watershed in optimization isn’t between linearity and nonlinearity but convexity and nonconvexity.”