Math 531A/532A: Variational Analysis
Terry Rockafellar

Autumn/Winter 1999-2000, MWF 8:30-9:20

The study of "variational problems" goes back to the first attempts, about 300 years ago, to determine curves or surfaces that are optimal according to some criterion.  Variations enter the picture when comparing a candidate curve or surface to its neighbors within the constraints that are imposed.  In modern times a vast range of other problems come under this heading, both finite-dimensional and infinite-dimensional, all of them focused on characterizing when a maximum or minimum is attained and what might happen to it under perturbations.

A central issue is how to exploit in such a characterization the structure of the function being maximized or minimized and of the set over which this is to take place.  In the classical mindset, the function would in some way be differentiable and the set would be given by equations, likewise in terms of differentiable functions.  Nowadays, though, it is apparent that only a relatively limited class of applications fits that simple picture.   Tremendous effort has gone into developing a better mathematical framework that reaches far beyond the inherited ideas of calculus, for example in replacing derivatives by "subderivatives."

This course will be devoted to explaining this modern form of variational analysis and its motivations, as well as to showing how it extends our understanding even of traditional problem types.

Plan for MATH 531A: The course will begin with a look at various classical and modern problems and the mathematical challenges they present.  It will then get down to the business of building up basic tools for handling the operations of max and min.  A major theme will be the variational geometry of sets (generalized tangent and normal vectors) and its application to the epigraphs of functions.  Epigraphs take over the role usually assigned to graphs because the functions that have to be treated are often extended-real-valued.  This application of geometric notions leads to a fascinating and powerful "subdifferential calculus."

Most of the work that will be undertaken in 531A will be in a finite-dimensional framework and will draw on the recently published book Variational Analysis (Rockafellar and Wets).  That book will act more or less as a text, but with considerable selectivity.

For 531A, students mainly need to have a good background in real analysis at least on the 400 level.  It would be helpful also to have taken the 515 course (Fundamentals of Optimization) for the sake of a broader appreciation of the aims and achievements of the subject.  The elements of variational analysis included in 515 will, however, be covered in 531A independently in more advanced form.

Plan for MATH 532A: The course will move on from the groundwork laid in the autumn to the study of problems in the mode of the "calculus of variations" but with broader structure, admitting inequality constraints and encompassing models in optimal control.  In these problems, integral functionals are the key, and a major complication is that the integrands in these functionals can be extended-real-valued.  Subdifferential calculus nonetheless makes it possible to derive optimality conditions that generalize the famed Euler-Lagrange equation and its Hamiltonian counterpart, for instance.

Students in 532A will have to be ready to work in standard function spaces and to cope with questions of measurability.  The theory of measurable selections from set-valued mappings will be developed and put to much use in this endeavor.