Subgradients and Variational Analysis

The study of problems of maximization or minimization subject to constraints has been a fertile field for the development of mathematical analysis from classical times. In recent decades, convexity has come forward as an important tool, and the geometry of convexity has been translated into notions of directional derivatives and subgradients of functions that may not be differentiable in the ordinary sense. Now there has emerged a form of analysis able to deal robustly, even in the absence of convexity, with the phenomena of nonsmoothness that arise in variational problems. The Need for Variational Analysis Differential calculus has been so successful in treating a variety of physical phenomena that mathematics has long relied on differentiable functions as the main tools of analysis. The domain of such a function, in a finite-dimensional setting, is typically an open subset of IR or of a differentiable manifold such as might be defined by a system of equations in IR and coordinatized locally by IR for some d < n. Many of the systems and phenomena that have come under mathematical scrutiny in recent decades, however, are of interest especially at their frontiers of feasibility. They involve functions and mappings whose domains may be closed sets with very complicated boundaries, expressed often by numerous inequalities as well as equations. Behavior around boundary points of these domains is seen as crucial, but it cannot well be investigated without a development of ideas beyond the customary framework. A major source of this trend lies in the fact that mathematical models are being used more and more for prescriptive as well as merely descriptive purposes. Nowadays one seeks not only to describe what happens in the world but to influence or improve the way it happens. New subjects have been created like optimization theory, control theory, and viability theory, which are heavily involved with finding the extremes of what may be possible under given circumstances. This has stemmed from an increasing preoccupation ∗ The written text of this lecture has been adapted from the introduction of the forthcoming book Variational Analysis by R. T. Rockafellar and R. J-B Wets.