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 became 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.
mathematics as thought process
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.
It’s the excitement of discovering new properties and relationships—ones having the intellectual beauty that only mathematics seems able to bring—that keeps me going. I never get tired of it. This process builds its own momentum. New flashes of insight stimulate curiosity more and more.
Of course, a mathematician has to be in tune with some of the basics of a mathematical way of life, such as pleasure in spending hours in quiet contemplation, and in dedication to writing projects. But we all know that this somewhat solitary side of mathematical life also brings with it a kind of social life that few people outside of our professional world can even imagine. The frequent travel that’s not just tied to a few laboratories, the network of friends and research collaborators in different cities and even different countries, the extended family of former students, and the interactions with current students—what fun, and what an opportunity to explore music, art, nature, and our many other interests. All these features keep me going too.
Recently I was asked whether I really liked mountain hiking and backpacking. The interviewer had seen that about me on a web site and appeared to be incredulous that someone with such outdoor activities could fit her mental picture of a mathematician. So little did she know about the lives we lead!
too many journals
There are too many meetings nowadays, even too many in some specialized areas of Optimization. This is regrettable, but perhaps self-limiting because of constraints on the time and budgets of participants.
An aspect of meetings that I believe can definitely have a bad effect on the quality of publications is the proliferation of conference volumes of collected papers. This isn’t a new thing, but has gotten worse. In principle such volumes could be good, but we all know that it’s not a good idea to submit a real paper to such a volume. In fact I often did that in the past, but it’s clear now that such papers are essentially lost to the literature after a few years and unavailable.
There are also too many journals. This is a difficult matter, but it may also be self-limiting. Many libraries now aren’t subscribing to all the available journals. At my own university, for example, we have decided to omit many mathematical journals that we regard as costing much more than they are worth, and this even includes some older journals that are quite well known (I won’t name names). And hardly a month goes by without the introduction of yet another journal. Besides the problem of paying for all the journals (isn’t this often really a kind of business trick of publishers in which ambitious professors cooperate?), there is the quality problem that there aren’t enough researchers to referee the papers that get submitted. Furthermore, one sees that certain fields of research, that are perhaps questionable in value and content, start separate journals of their own and thereby escape their critics on the outside.
That book took over 10 years to write—if one includes the fact that at least twice we decided to start the job from the beginning again, totally reorganizing what we had. In that period I had the feeling of an enormous responsibility, but a joyful burden one even if involved with pain, somewhat like a woman carrying a baby within her and finally giving birth. I am very happy with the book (although it would be nice to have an opportunity to make a few little corrections), and Wets and I have heard many heart-warming comments about it.
Still, I have to confess that I have gone through a bit of post partum depression since it was finished. It’s clear—and we knew it always —that such a massive amount of theory can’t be digested very quickly, even by those who could benefit from it the most. Another feature of the situation, equally predictable, is that some of the colleagues who could most readily understand what we have tried to do often have their own philosophies and paradigms to sell. It’s discouraging to run into circumstances where developments we were especially proud of, and which we regarded as very helpful and definitive, appear simply to be ignored.
But in all this I have a very long view. We now take for granted that convex analysis is a good subject with worthwhile ideas, yet it was not always that way. There was actually a lot of resistance to it in the early days, from individuals who preferred a geometric presentation to one targeting concepts of analysis. Even on the practical plane, it’s fair to say that little respect was paid to convex analysis in numerical optimization until around 1990, say. Having seen how ideas that are vital, and sound, can slowly win new converts over many years, I can well dream that the same will happen with variational analysis. — R. Tyrrell Rockafellar