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R. Tyrrell Rockafellar

R. Tyrrell Rockafellar, ca.2009
R. Tyrrell Rockafellar, ca.2009

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 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.

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