Variational Principles

Abstract

Consider a C ¹ function $f\,:\,\mathbb{R}^n \to \mathbb{R}.$ If f has a (global) minimizer $\overline x$, then we know that $\nabla \,f\left( {\overline x } \right)\, = 0.$ Suppose, however, that f is bounded below, but does not have a minimizer. Is it possible to find a minimizing sequence $\left\{ {x_n } \right\}$ satisfying $f(x_n )\to$ inf f that also satisfies $\nabla f(x_n )\, \to \,0?$ If this is true, then it should be possible to obtain new optimality conditions even when minimizers do not exist. A celebrated result of Ekeland [86] known as Ekeland’s є-variational principle ensures that the answer to the above question is yes. Moreover, this principle is valid in a much more general context and has turned out to be one of the most important of the recent contributions to analysis. In this chapter, we give a fairly detailed exposition of this important principle and some of its applications. These include a short proof of Banach’s fixed point theorem, a characterization of the consistency of a system of linear inequalities, and the proofs of some of the most basic theorems of analysis (the open mapping theorem, Graves’s theorem, Lyusternik’s theorem, the inverse function theorem, and the implicit function theorem). Another significant application to the derivation of the Fritz John conditions in nonlinear programming is postponed to Section 9.3.