Mixed-Integer Nonlinear Optimization (MINLP) is the mother of all (deterministic) optimization problems. As such, it is too broad a category to do something for in general. Still, there are broad classes where we have positive complexity results, as well as narrow classes where we have negative complexity results. I will sample some of the landscape.
Considering very damning complexity results, we should have modest goals on the computational side. Nonetheless, there has been some substantial success with “general-purpose” algorithms and software for MINLP, applicable to rather broad (and practically relevant) formulations. In particular, for formulations with convex relaxations we have Bonmin (which implements NLP-based branch-and-bound and outer approximation), and for “factorable” formulations with non-convex relaxations we have Baron, Scip, Antigone, and Couenne (which implement spatial branch-and-bound). Still, the situation is not that we have extremely reliable nor scalable technology (in sharp contrast with LP and in less sharp contrast to MILP). Much of the research effort in modeling and algorithm improvement relates to finding tight convexifications. I will present recent mathematical efforts that I have been involved in to make algorithmic approaches to MILP more efficient and robust. In doing so, many areas of mathematics surprisingly pop up. Substantial parts of this are joint works with Daphne Skipper (U.S. Naval Academy) and with Emily Speakman (U. Michigan).
Jon Lee, University of Michigan
Jon Lee is the G. Lawton and Louise G. Johnson Professor of Engineering at the University of Michigan. He received his Ph.D. from Cornell University. Jon has previously been a faculty member at Yale University and the University of Kentucky, and an adjunct professor at New York University. He was a Research Staff member at the IBM T.J. Watson Research Center, where he managed the mathematical programming group. Jon is author of ~120 papers, the text “A First Course in Combinatorial Optimization” (Cambridge University Press), and the open-source book “A First Course in Linear Optimization” (Reex Press). He was the founding Managing Editor of the journal Discrete Optimization (2004-06), he is currently co-Editor of the journal Mathematical Programming, and he is on the editorial boards of the journals Optimization and Engineering, and Discrete Applied Mathematics. Jon was Chair of the Executive Committee of the Mathematical Optimization Society (2008-10), and Chair of the INFORMS Optimization Society (2010-12). He was awarded the INFORMS Computing Society Prize (2010), and he is a Fellow of INFORMS (since 2013).