Imre Pólik Mixed-integer second-order cone optimization (MISOCO) has become very popular in the last decade. Various aspects of solving these problems in Branch and Conic Cut (BCC) algorithms have been studied in the literature. This study aims to fill a gap and provide a novel way to find feasible solutions early in the BCC algorithm. Such solutions … Read more
We present a new branch-and-bound type search method for mixed integer linear optimization problems based on the concept of offshoots (introduced in this paper). While similar to a classic branch-and-bound method, it allows for changing the order of the variables in a dive (shaping) and removing unnecessary branching variables from a dive (trimming). The regular … Read more
Interior point methods (IPM) are the most popular approaches to solve Second Order Cone Optimization (SOCO) problems, due to their theoretical polynomial complexity and practical performance. In this paper, we present a warm-start method for primal-dual IPMs to reduce the number of IPM steps needed to solve SOCO problems that appear in a Branch and … Read more
We study the convex hull of the intersection of a disjunctive set defined by parallel hyperplanes and the feasible set of a mixed integer second order cone optimization problem. We extend our prior work on disjunctive conic cuts, which has thus far been restricted to the case in which the intersection of the hyperplanes and … Read more
We present two novel applications of symmetries for mixed-integer linear programming. First we propose two variants of a new heuristic to improve the objective value of a feasible solution using symmetries. These heuristics can use either the actual permutations or the orbits of the variables to find better feasible solutions. Then we introduce a new … Read more
We investigate families of quadrics that have fixed intersections with two given hyper-planes. The cases when the two hyperplanes are parallel and when they are nonparallel are discussed. We show that these families can be described with only one parameter. In particular we show how the quadrics are transformed as the parameter changes. This research … Read more
We study the convex hull of the intersection of a convex set E and a linear disjunction. This intersection is at the core of solution techniques for Mixed Integer Conic Optimization. We prove that if there exists a cone K (resp., a cylinder C) that has the same intersection with the boundary of the disjunction … Read more
The following question arises in stochastic programming: how can one approximate a noisy convex function with a convex quadratic function that is optimal in some sense. Using several approaches for constructing convex approximations we present some optimization models yielding convex quadratic regressions that are optimal approximations in $L_1$, $L_\infty$ and $L_2$ norm. Extensive numerical experiments … Read more
We consider the complexity of gap-free duals in semidefinite programming. Using the theory of homogeneous cones we provide a new representation of Ramana’s gap-free dual and show that the new formulation has a much better complexity than originally proved by Ramana. CitationCOR@L Technical Report, Lehigh UniversityArticleDownload View PDF
We present a strong duality theory for optimization problems over symmetric cones without assuming any constraint qualification. We show important complexity implications of the result to semidefinite and second order conic optimization. The result is an application of Borwein and Wolkowicz’s facial reduction procedure to express the minimal cone. We use Pataki’s simplified analysis and … Read more