In this article we introduce an inner parallel cutting plane method (IPCP) to compute good feasible points along with valid cutting planes for mixed-integer convex optimization problems. The method iteratively generates polyhedral outer approximations of an enlarged inner parallel set (EIPS) of the continuously relaxed feasible set. This EIPS possesses the crucial property that any … Read more
Semidefinite programming relaxations complement polyhedral relaxations for quadratic optimization, but global optimization solvers built on polyhedral relaxations cannot fully exploit this advantage. This paper develops linear outer-approximations of semidefinite constraints that can be effectively integrated into global solvers. The difference from previous work is that our proposed cuts are (i) sparser with respect to the … Read more
In this self-contained paper, we present a theory of the piecewise linear minimal valid functions for the 1-row Gomory-Johnson infinite group problem. The non-extreme minimal valid functions are those that admit effective perturbations. We give a precise description of the space of these perturbations as a direct sum of certain finite- and infinite-dimensional subspaces. The … Read more
We propose a novel way of applying cutting plane techniques to two-stage mixed-integer stochastic programs. Instead of using cutting planes that are always valid, our idea is to apply inexact cutting planes to the second-stage feasible regions that may cut away feasible integer second-stage solutions for some scenarios and may be overly conservative for others. … Read more
This paper is concerned with a cutting-plane algorithm for solving mixed-integer semidefinite optimization (MISDO) problems. In this algorithm, the positive semidefinite constraint is relaxed, and the resultant mixed-integer linear optimization problem is repeatedly solved with valid inequalities for the relaxed constraint. We prove convergence properties of the algorithm. Moreover, to speed up the computation, we … Read more
Given a point x inside a polytope P and a direction d, the projection of x along d asks to find the maximum step length t such that x+td is feasible; we say x+td is a pierce point because it belongs to the boundary of P. We address this projection sub-problem with arbitrary interior points … Read more
We consider the integer points in a unimodular cone K ordered by a lexicographic rule defined by a lattice basis. To each integer point x in K we associate a family of inequalities (lex-inequalities) that defines the convex hull of the integer points in K that are not lexicographically smaller than x. The family of … Read more
In this paper, an open source solver for mixed-integer nonlinear programming (MINLP) problems is presented. The Supporting Hyperplane Optimization Toolkit (SHOT) combines a dual strategy based on polyhedral outer approximations (POA) with primal heuristics. The outer approximation is achieved by expressing the nonlinear feasible set of the MINLP problem with linearizations obtained with the extended … Read more
As an alternative to traditional integer programming (IP), decision diagrams (DDs) provide a new solution technology for discrete problems based on their combinatorial structure and dynamic programming representation. While the literature mainly focuses on the competitive aspects of DDs as a stand-alone solver, we investigate their complementary role by studying IP techniques that can be … Read more
Here we present a center-cut algorithm for convex mixed-integer nonlinear programming (MINLP) that can either be used as a primal heuristic or as a deterministic solution technique. Like many other algorithms for convex MINLP, the center-cut algorithm constructs a linear approximation of the original problem. The main idea of the algorithm is to use the … Read more