Recently, mixed-integer programming (MIP) techniques have been applied to learn optimal decision trees. Empirical research has shown that optimal trees typically have better out-of-sample performance than heuristic approaches such as CART. However, the underlying MIP formulations often suffer from slow runtimes, due to weak linear programming (LP) relaxations. In this paper, we first propose a … Read more
We consider the convex quadratic optimization problem with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic number of additional variables consists of a single positive semidefinite constraint (explicitly stated) and linear constraints. In particular, convexification of … Read more
The SCIP Optimization Suite provides a collection of software packages for mathematical optimization centered around the constraint integer programming framework SCIP. This paper discusses enhancements and extensions contained in version 8.0 of the SCIP Optimization Suite. Major updates in SCIP include improvements in symmetry handling and decomposition algorithms, new cutting planes, a new plugin type … Read more
The thermal unit commitment (UC) problem has historically been formulated as a mixed integer quadratic programming (MIQP), which is difficult to solve efficiently, especially for large-scale systems. The tighter characteristic reduces the search space, therefore, as a natural consequence, significantly reduces the computational burden. In literatures, many tightened formulations for a single unit with parts … Read more
We consider a general mixed-integer convex program. We first develop an algorithm for solving this problem, and show its nite convergence. We then develop a finitely convergent decomposition algorithm that separates binary variables from integer and continuous variables. The integer and continuous variables are treated as second stage variables. An oracle for generating a parametric … Read more
We show that there exists a family of instances of the lot-sizing problem, such that any branch-and-bound tree that solves them requires an exponential number of nodes, even in the case when the branchings are performed on general split disjunctions. ArticleDownload View PDF
The Total Matching Polytope generalizes the Stable Set Polytope and the Matching Polytope. In this paper, we give the perfect formulation for Trees and we derive two new families of valid inequalities, the balanced biclique inequalities which are always facet-defining and the non-balanced lifted biclique inequalities obtained by a lifting procedure, which are facet-defining for … Read more
Given a grid of active and inactive pixels, the weighted rectangles partitioning (WRP) problem is to find a maximum-weight partition of the active pixels into rectangles. WRP is formulated as an integer programming problem and instances with an integral relaxation polyhedron are characterized by a balanced problem matrix. A complete characterization of these balanced instances … Read more
We study a delivery strategy for last-mile deliveries in urban areas which combines freight transportation with mass mobility systems with the goal of creating synergies contrasting negative externalities caused by transportation. The idea is to use the residual capacity on public transport means for moving freights within the city. In particular, the system is such … Read more
We consider nonlinear optimization problems that involve surrogate models represented by neural net-works. We demonstrate first how to directly embed neural network evaluation into optimization models, highlight a difficulty with this approach that can prevent convergence, and then characterize stationarity of such models. We then present two alternative formulations of these problems in the specific … Read more