Convex Optimization - last lecture at Stanford

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Stephen Boyd & Brad Osgood (right) behind Packard building (amidst construction) prior to Stanford 2009 Commencement

In 1999, Prof. Stephen Boyd's class on Convex Optimization required no textbook; just his lecture notes and figures drawn freehand. Boyd said there were about 100 people in the world who understood the topic. At the time of his first lecture in Spring 2009, that number of people had risen to 1000 by his estimation.

It was fascinating to witness evolution of this Course at Stanford over that ten year period, but more fascinating were the last five minutes of the last class in Spring 2009.

Perhaps because the lecture was not videotaped, Boyd revealed more about his professional experiences than he may have otherwise.

Certainly, this was something he had not shared throughout the preceding lectures which were quite academic, in the best sense:

Revelation

Some of Boyd's colleagues and contemporaries "don't believe in" Convex Optimization.

Always light in his presentation, Boyd recounted several incidents over his career in quite a humorous entertaining manner: Essentially, a colleague would ask for help solving a problem; say, in antenna design. Boyd would provide optimal solution to the problem, but then that colleague would reply by pointing out its weaknesses. The colleague might say something like "Everyone knows that those coefficients can't be negative because the antenna will wiggle uncontrollably." Inevitably, Boyd would respond by reminding that colleague: there was no previous mention of such a constraint. The new constraint would be accounted for, the problem solved again, and then another round of the same would ensue.

Another colleague proposed that in some circumstances "One does not want an optimal solution";  implying, optimal solutions are bad in some sense.

Yet another esteemed colleague posited an optimal solution provided by Boyd as proof that "Convex Optimization doesn't work."

Boyd ended the lecture with the moral of his recounting (which I paraphrase): If constraints are incomplete, then an optimal solution cannot be right. The most difficult part of all is to express a problem well.

Conclusion

Stephen Boyd did not invent Convex Optimization, but he probably deserves much of the credit for its popularization in engineering. His book with Vandenberghe Convex Optimization (2004) certainly made this emerging field of mathematics more accessible.

If they don't believe Stephen Boyd, who then?

The beauty of mathematics is that either it is right or it is wrong; e.g., the left side of an equation equals the right side. There is little room for interpretation as there may be in other disciplines (e.g., Law). Mathematical results are traditionally presented within a theorem/proof paradigm. A proof represents culmination of labor that can span many years; indeed, Fermat's last theorem took more than 300 years. Physicist Richard Feynman calls trivial any proven theorem, but not to diminish the result.

So we may interpret disbelief in Convex Optimization as disbelief in the proofs; not disproof, just plain disbelief. What is disquieting about Boyd's revelation is that these particular colleagues are intelligent, successful, and learned people from academia and industry.