We study the integrality gap of convex mixed-integer programs, that is, the difference between the optimal value of such a problem and the optimal value of its continuous relaxation. We study classes of convex sets whose associated optimization problem have finite integrality gap: Dirichlet convex sets, sets with full-dimensional recession cones and sets that can be approximated by polyhedral sets. In the latter two cases, we provide estimates for the value of the integrality gap. Finally, we study the possibility of estimating the integrality gap of nonlinear convex mixed-integer programs via rational polyhedral approximations of their feasible regions and argue that, in general, such an approach may yield arbitrarily worse bounds compared to integrality gap estimations specifically derived by studying the associated nonlinear set. Based on joint work with Burak Kocuk and Silvia Di Gregorio.
Diego Moran Ramirez is an assistant professor in the Department of Industrial and Systems Engineering at Rensselaer Polytechnic Institute. He holds a Ph.D. in Operations Research from the Georgia Institute of Technology, a Mathematical Engineer professional degree (BS and MS equivalent) and a Master in Operations Management from Universidad de Chile. Prior to joining RPI, Dr. Moran Ramirez was an assistant professor in the School of Business at Universidad Adolfo Ibañez and an assistant professor in the Industrial and Systems Engineering department at Virginia Tech. His main research interest is Optimization with a focus on the theory and applications of mixed-integer linear and nonlinear programming. His research in mathematical optimization is motivated by algorithmic aspects of mixed-integer programs, and he has also worked on projects involving the application of optimization to solve real-world problems arising in production planning, logistics and other relevant business and engineering areas. He received the INFORMS Optimization Society Student Paper Prize in 2012 and the IBM Faculty Award in 2018.
