Andres Gomez

Research Summary

The past two decades have witnessed an explosion in the use of optimization to tackle problems arising in data analysis, finance, statistics and, more recently, machine learning. These new application domains demand faster, scalable and more precise algorithms, yet classical optimization tools and techniques have been unable to cope with such requirements. Specifically, there has been an enormous progress in solving convex optimization problems. However, most inference problems are naturally non-convex once priors or interpretability conditions are imposed. Thus, decision-makers are faced with a dichotomy: either construct a (crude) convex approximation of the problem, which can be solved efficiently but yields sub-optimal and even bad solutions; alternatively, tackle the non-convex directly to obtain optimal solutions, but at the expense of large or excessive computational times.

Dr. Gómez goal is to bridge the gap between these two extremes. His research focuses on systematically constructing strong or ideal convex relaxations of difficult problems. Such relaxations can then be naturally used to obtain high-quality solutions quickly, and to solve the problems to optimality efficiently, resulting in the best of both worlds. His research uses ideas from the following disciplines:
- Discrete optimization (combinatorial, submodularity)
- Mixed-integer optimization (branch-and-bound, lifting, disjunctive programming)
- Convex optimization (quadratic, conic).