MemComputing Integer Linear Programming

Integer linear programming (ILP) encompasses a very important class of optimization problems that are of great interest to both academia and industry. Several algorithms are available that attempt to explore the solution space of this class efficiently, while requiring a reasonable compute time. However, although these algorithms have reached various degrees of success over the years, they still face considerable challenges when confronted with particularly hard problem instances, such as those of the MIPLIB 2010 library. In this work we propose a radically different non-algorithmic approach to ILP based on a novel physics-inspired computing paradigm: Memcomputing. This paradigm is based on digital (hence scalable) machines represented by appropriate electrical circuits with memory. These machines can be either built in hardware or, as we do here, their equations of motion can be efficiently simulated on our traditional computers. We first describe a new circuit architecture of memcomputing machines specifically designed to solve for the linear inequalities representing a general ILP problem. We call these self-organizing algebraic circuits, since they self-organize dynamically to satisfy the correct (algebraic) linear inequalities. We then show simulations of these machines using MATLAB running on a single core of a Xeon processor for several ILP benchmark problems taken from the MIPLIB 2010 library, and compare our results against a renowned commercial solver. We show that our approach is very efficient when dealing with these hard problems. In particular, we find within minutes feasible solutions for one of these hard problems (f2000 from MIPLIB 2010) whose feasibility, to the best of our knowledge, has remained unknown for the past eight years

Parameterizing Branch-and-Bound Search Trees to Learn Branching Policies

Branch and Bound (B&B) is the exact tree search method typically used to solve Mixed-Integer Linear Programming problems (MILPs). Learning branching policies for MILP has become an active research area, with most works proposing to imitate the strong branching rule and specialize it to distinct classes of problems. We aim instead at learning a policy that generalizes across heterogeneous MILPs: our main hypothesis is that parameterizing the state of the B&B search tree can aid this type of generalization. We propose a novel imitation learning framework, and introduce new input features and architectures to represent branching. Experiments on MILP benchmark instances clearly show the advantages of incorporating an explicit parameterization of the state of the search tree to modulate the branching decisions, in terms of both higher accuracy and smaller B&B trees. The resulting policies significantly outperform the current state-of-the-art method for "learning to branch" by effectively allowing generalization to generic unseen instances.Comment: AAAI 2021 camera-ready version with supplementary materials, improved readability of figures in main article. Code, data and trained models are available at https://github.com/ds4dm/branch-search-tree

Exploring core points for fun and profit: A study of lattice-free orbit polytopes

This thesis studies minimal lattice-free symmetric polytopes. Lattice-free means that the only integral points in the polytope are its vertices. Symmetric in context of the thesis means that all vertices lie in one single orbit under a group action. The thesis focuses on groups that are permutation groups acting on R^n by permuting coordinates. If a symmetric polytope is lattice-free, its vertices are called core points. Methods to construct core points and applications in symmetric integer linear programming are explored.Diese Arbeit behandelt gitterpunkt-freie symmetrische Polytope. Gitterpunkt-frei heißt, dass die Ecken des Polytops die einzigen enthaltenen ganzzahligen Punkte sind. Symmetrisch im Kontext dieser Arbeit meint, dass alle Ecken in einem einzigen Orbit einer Gruppenwirkung liegen. Diese Arbeit beschäftigt sich besonders mit Gruppen, die als Permutationsgruppen auf R^n wirken, indem sie Koordinaten permutieren. Die Ecken eines gitterpunkt-freien symmetrischen Polytops werden core points genannt. Es werden Methoden entwickelt, core points zu finden und in ganzzahliger Optimierung anzuwenden

The Chebyshev center as an alternative to the analytic center in the feasibility pump

© The Author(s) 2023As a heuristic for obtaining feasible points of mixed integer linear problems, the feasibility pump (FP) generates two sequences of points: one of feasible solutions for the relaxed linear problem; and another of integer points obtained by rounding the linear solutions. In a previous work, the present authors proposed a variant of FP, named analytic center FP, which obtains integer solutions by rounding points in the segment between the linear solution and the analytic center of the polyhedron of the relaxed problem. This work introduces a new FP variant that replaces the analytic center with the Chebyshev center. Two of the benefts of using the Chebyshev center are: (i) it requires the solution of a linear optimization problem (unlike the analytic center, which involves a convex nonlinear optimization problem for its exact solution); and (ii) it is invariant to redundant constraints (unlike the analytic center, which may not be well centered within the polyhedron for problems with highly rank-defcient matrices). The computational results obtained with a set of more than 200 MIPLIB2003 and MIPLIB2010 instances show that the Chebyshev center FP is competitive and can serve as an alternative to other FP variants.This research has been supported by the MCIN/AEI/FEDER project RTI2018-097580-B-I00. Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer NaturePeer ReviewedPostprint (published version