Orbital shrinking

Symmetry plays an important role in optimization. The usual approach to cope with symmetry in discrete optimization is to try to eliminate it by introducing artificial symmetry-breaking conditions into the problem, and/or by using an ad-hoc search strategy. In this paper we argue that symmetry is instead a beneficial feature that we should preserve and … Read more

A short note on the global convergence of the unmodified PRP method

It is well-known that the search direction generated by the standard (unmodified) PRP nonlinear conjugate gradient method is not necessarily a descent direction of the objective function, which brings difficulty for its global convergence for general functions. However, to our surprise, it is easily proved in this short note that the unmodified PRP method still … Read more

COIN-OR METSlib: a Metaheuristics Framework in Modern C++.

The document describes COIN-OR METSlib, a C++ framework for local search based metaheuristics. METSlib has been used to implement a massively parallel VRP algorithm, a state of the art Vertex Coloring Problem solver, a Timetabling software, and in many other projects. Article Download View COIN-OR METSlib: a Metaheuristics Framework in Modern C++.

PuLP: A Linear Programming Toolkit for Python

This paper introduces the PuLP library, an open source package that allows mathematical programs to be described in the Python computer programming language. PuLP is a high-level modelling library that leverages the power of the Python language and allows the user to create programs using expressions that are natural to the Python language, avoiding special … Read more

An LP-Newton Method: Nonsmooth Equations, KKT Systems, and Nonisolated Solutions

We define a new Newton-type method for the solution of constrained systems of equations and analyze in detail its properties. Under suitable conditions, that do not include differentiability or local uniqueness of solutions, the method converges locally quadratically to a solution of the system of equations, thus filling an important gap in the existing theory. … Read more

A First Order Method for Finding Minimal Norm-Like Solutions of Convex Optimization Problems

We consider a general class of convex optimization problems in which one seeks to minimize a strongly convex function over a closed and convex set which is by itself an optimal set of another convex problem. We introduce a gradient-based method, called the minimal norm gradient method, for solving this class of problems, and establish … Read more

Lattice-free sets, multi-branch split disjunctions, and mixed-integer programming

In this paper we study the relationship between valid inequalities for mixed-integer sets, lattice-free sets associated with these inequalities and the multi-branch split cuts introduced by Li and Richard (2008). By analyzing $n$-dimensional lattice-free sets, we prove that for every integer $n$ there exists a positive integer $t$ such that every facet-defining inequality of the … Read more

Robust Network Design: Formulations, Valid Inequalities, and Computations

Traffic in communication networks fluctuates heavily over time. Thus, to avoid capacity bottlenecks, operators highly overestimate the traffic volume during network planning. In this paper we consider telecommunication network design under traffic uncertainty, adapting the robust optimization approach of Bertsimas and Sim (2004). We present three different mathematical formulations for this problem, provide valid inequalities, … Read more

A (k+1)-Slope Theorem for the k-Dimensional Infinite Group Relaxation

We prove that any minimal valid function for the k-dimensional infinite group relaxation that is piecewise linear with at most k+1 slopes and does not factor through a linear map with non-trivial kernel is extreme. This generalizes a theorem of Gomory and Johnson for k=1, and Cornu\’ejols and Molinaro for k=2. Article Download View A … Read more