# Rockafellar

### From Wikimization

# R. Tyrrell Rockafellar

*Some of the history of *convex analysis* is recounted in the notes at the ends of the first two chapters of my book Variational Analysis, written with Roger Wets. Before the early 1960’s, there was plenty of convexity, but almost entirely in geometric form with little that could be called *analysis*. The geometry of convex sets had been studied by many excellent mathematicians, *e.g.* Minkowski, and had become important in functional analysis, specifically in Banach space theory and the study of norms. Convex functions other than norms began to attract much more attention once optimization started up in the early 1950’s, and through the economic models that became popular in the same era, involving games, utility functions, and the like. Still, convex functions weren’t handled in a way that was significantly different from that of other functions. That only came to be true later.*

*As a graduate student at Harvard, I got interested in convexity because I was amazed by linear programming duality and wanted to invent a *nonlinear programming duality.* That was around 1961. The excitement then came from all the work going on in optimization, as represented in particular by the early volumes of collected papers being put together by Tucker and others at Princeton, and from the beginnings of what later become the sequence of Mathematical Programming Symposia. It didn’t come from anything in convexity itself. At that time, I knew of no one else who was really much interested in trying to do *new* things with convexity. Indeed, nobody else at Harvard had much awareness of convexity, not to speak of Optimization.*

*It was while I was writing up my dissertation—focused then on dual problems stated in terms of polar cones—that I came across Fenchel’s conjugate convex functions, as described in Karlin’s book on game theory. They turned out to be a wonderful vehicle for expressing *nonlinear programming duality,* and I adopted them wholeheartedly. Around the time the thesis was nearly finished, I also found out about Moreau’s efforts to apply convexity ideas, including duality, to problems in mechanics.*

*Moreau and I independently in those days at first, but soon in close exchanges with each other, made the crucial changes in outlook which, I believe, created *convex analysis* out of *convexity*. For instance, he and I passed from the basic objects in Fenchel’s work, which were pairs consisting of a convex set and a finite convex function on that set, to extended-real-valued functions implicitly having *effective domains,* for which we moreover introduced set-valued subgradient mappings. Nevertheless, the idea that convex functions ought to be treated geometrically in terms of their epigraphs instead of their graphs was essentially something we had gotten from Fenchel.*

*I went to Princeton for a whole academic year through an invitation from Tucker. I had kept contact with him as a student, even though I was at Harvard, not Princeton, and had never actually met him. (He had helped to convince my advisor that my research was promising.) He had me teach a course on convex functions, for which I wrote the lecture notes, and he then suggested that those notes be expanded to a book. And yes, it was he who suggested the title, *Convex Analysis*, thereby inventing the name for the new subject.*

*I think of Klee (a long-time colleague of mine in Seattle, who helped me get a job there), and Valentine (whom I once met but only briefly), as well as Caratheodory, as involved with *convexity* rather than *convex analysis*. Their contributions can be seen as primarily geometric.*

*Ideally, mathematics should be seen as a thought process, rather than just as a mass of facts to be learned and remembered, which*
is so often the common view.

*Even with standard subjects such as calculus, I think it’s valuable to communicate the excitement of the ideas and their history, how hard they were to develop and understand properly—which so often reflects difficulties that students have themselves.*