More than 50 years of development have made mixed integer linear
programming (MILP) an extremely successful tool. MILP’s modeling
flexibility allows it describe a wide range of business, engineering
and scientific problems, and, while MILP is NP-hard, many of these
problems are routinely solved in practice thanks to state-of-the-art
solvers that nearly double their machine-independent speeds every
year. Inspired by this success, the last decade has seen a surge of
activity on the solution and application of mixed integer convex
programming (MICP), which extends MILP’s versatility by allowing the
use of convex constraints in addition to linear inequalities. In this
talk we cover various recent developments concerning theory,
algorithms and computation for MICP. Solvers for MICP can be
significantly more effective than those for more general non-convex
optimization, so one of the questions we cover in this talk is what
classes of non-convex constraints can be modeled through MICP. We also
cover various topics concerning the modeling and computational
solution of MICP problems using the Julia programming language and the
JuMP modeling language for optimization. In particular, we show how
mixed integer optimal control problems where the variables are
polynomials can be easily modeled and solved by seamlessly combining
several Julia packages and JuMP extensions with the Julia-written MICP
solver Pajarito.jl. Finally, we introduce the Julia-based interior
point solver for general non-symmetric cones Hypatia.jl, and show how
its use of non-standard cones can allow it to outperform commercial
solvers in certain instances.

Bio

Juan Pablo Vielma is the Richard S. Leghorn (1939) Career Development
Associate Professor at MIT Sloan School of Management and is
affiliated to MIT’s Operations Research Center. Dr. Vielma has a B.S.
in Mathematical Engineering from University of Chile and a Ph.D. in
Industrial Engineering from the Georgia Institute of Technology. His
current research interests include the theory and practice of
mixed-integer mathematical optimization and applications in energy,
natural resource management, marketing and statistics. In January of
2017 he was named by President Obama as one of the recipients of the
Presidential Early Career Award for Scientists and Engineers (PECASE).
Some of his other recognitions include the NSF CAREER Award and the
INFORMS Computing Society Prize. He is currently an associate editor
for Operations Research and Operations Research Letters, a member of
the board of directors of the INFORMS Computing Society, and a member
of the NumFocus steering committee for JuMP.