Now that I've returned to Socorro, it's a good time to sit down and reflect on my recent sabbatical leave at the Institute for Pure and Applied Mathematics (IPAM) at UCLA.
During this visit to IPAM I was a core participant in the Fall 2010 long program on Modern Trends In Optimization And Its Application. This meant that I was in residence at IPAM for 3 months and got to attend a series of week long workshops on various optimization topics. The other core participants were mainly graduate students and postdocs from UCLA, Cal Tech, Stanford, MIT, Rice, the University of Washington, and other places in the US, as well as international visitors from Belgium, France, Germany, Israel, Singapore, and the UK. In addition to the younger folks, there were a few of us more senior folks. I really enjoyed my time with this group of highly motivated and energetic optimization researchers.
The title of the long program was deliberately broad, but as you'll see in a moment, nearly all of the research presented in the various workshops related to convex optimization and applications of convex optimization.
The workshops included:
Optimization Tutorials. This first workshop consisted entirely of tutorial lectures introducing topics that appeared in later workshops.
Workshop I: Convex Optimization and Algebraic Geometry. This workshop focused on applications of convex optimization to polynomial optimization problems.
Workshop II: Numerical Methods for Continuous Optimization. This workshop focused primarily on first methods for convex optimization problems and particularly methods for problems involving nonsmooth convex functions that appear frequently in image processing and compressive sensing.
Workshop III: Discrete Optimization. Discrete optimization problems are inherently nonconvex. However, an important theme of this workshop was convex relaxations of discrete optimization problems, particularly SDP relaxations. There were also a couple of interesting talks related to graph sparsification.
Workshop IV: Robust Optimization. In robust optimization, we consider an optimization problem with problem data that are uncertain within a specified range or uncertainty set and formulate a robust counterpart to the original problem whose solution will be optimal regardless of where the problem data are within this uncertainty set. In most cases of interest the resulting reformulated problems are second order cone programs or semidefinite programming problems.
Workshop V: Applications of Optimization in Science and Engineering. This workshop focused mainly on applications of convex optimization in engineering design optimization and image processing.
Culminating Workshop at Lake Arrowhead. The culminating workshop at Lake Arrowhead was a chance for participants to present results of projects that they'd been working on during the long program. This relaxing week at UCLA's Lake Arrowhead conference center was a wonderful way to end the long program. By this time the group had bonded together, and this was our last chance to enjoy some social time together.
Overall, I would say that the workshops at IPAM give a very good overview of what's going on today in the area of convex optimization. Slides for many of the talks are available from the workshop web pages. In other cases you may have to contact the authors to get copies of the slides or related papers. I think that it would have been very helpful (particularly for the tutorial workshop) if IPAM had used a lecture capture system to capture talks so that others could see them.
In my next couple of postings I'll talk a bit more about some of the most important things that I learned during the long program.