11 research outputs found
Reformulating the Disjunctive Cut Generating Linear Program
Lift-and-project cuts can be obtained by defining an elegant optimization problem over the space of valid inequalities, the cut generating linear program (CGLP). A CGLP has two main ingredients: (i) an objective function, which invariably maximizes the violation with respect to a fractional solution x to be separated; and (ii) a normalization constraint, which limits the scale in which cuts are represented. One would expect that CGLP optima entail the best cuts, but the normalization may distort how cuts are compared, and the cutting plane may not be a supporting hyperplane with respect to the closure of valid inequalities from the CGLP. This work proposes the reverse polar CGLP (RP-CGLP), which switches the roles conventionally played by objective and normalization: violation with respect to x is fixed to a positive constant, whereas we minimize the slack for a point p that cannot be separated by the valid inequalities. Cuts from RP-CGLP optima define supporting hyperplanes of the immediate closure. When that closure is full-dimensional, the face defined by the cut lays on facets first intersected by a ray from x to p, all of which corresponding to cutting planes from RP-CGLP optima if p is an interior point. In fact, these are the cuts minimizing a ratio between the slack for p and the violation for x. We show how to derive such cuts directly from the simplex tableau in the case of split disjunctions and report experiments on adapting the CglLandP cut generator library for the RP-CGLP formulation
Benders decomposition for network design covering problems
Article number 105417We consider two covering variants of the network design problem. We are given a set of origin/destination
pairs, called O/D pairs, and each such O/D pair is covered if there exists a path in the network from the origin
to the destination whose length is not larger than a given threshold. In the first problem, called the Maximal
Covering Network Design problem, one must determine a network that maximizes the total fulfilled demand
of the covered O/D pairs subject to a budget constraint on the design costs of the network. In the second
problem, called the Partial Covering Network Design problem, the design cost is minimized while a lower
bound is set on the total demand covered. After presenting formulations, we develop a Benders decomposition
approach to solve the problems. Further, we consider several stabilization methods to determine Benders cuts
as well as the addition of cut-set inequalities to the master problem. We also consider the impact of adding
an initial solution to our methods. Computational experiments show the efficiency of these different aspects.Feder (UE) PID2019- 106205GB-I00FEDER(UE) MTM2015-67706-PFonds de la Recherche Scientifique PDR T0098.1
{\lambda}-Cent-Dians and Generalized-Center for Network Design
In this paper, we extend the notions of -cent-dian and
generalized-center from Facility Location Theory to the more intricate domain
of Network Design. Our focus is on the task of designing a sub-network within a
given underlying network while adhering to a budget constraint. This
sub-network is intended to efficiently serve a collection of origin/destination
pairs of demand. % rather than individual points.
The -cent-dian problem studies the balance between efficiency and
equity. We investigate the properties of the -cent-dian and
generalized-center solution networks under the lens of equity, efficiency, and
Pareto-optimality. We provide a mathematical formulation for
and discuss the bilevel structure of this problem for . Furthermore,
we describe a procedure to obtain a complete parametrization of the
Pareto-optimality set based on solving two mixed integer linear formulations by
introducing the concept of maximum -cent-dian. We evaluate the quality
of the different solution concepts using some inequality measures. Finally, for
, we study the implementation of a Benders decomposition
method to solve it at scale
Models and Algorithms for the Product Pricing with Single-Minded Customers Requesting Bundles
International audienceWe analyze a product pricing problem with single-minded customers, each interested in buying a bundle of products. The objective is to maximize the total revenue and we assume that supply is unlimited for all products. We contribute to a missing piece of literature by giving some mathematical formulations for this single-minded bundle pricing problem. We first present a mixed-integer nonlinear program with bilinear terms in the objective function and the constraints. By applying classical linearization techniques, we obtain two different mixed-integer linear programs. We then study the polyhedral structure of the linear formulations and obtain valid inequalities based on an RLT-like framework. We develop a Benders decomposition to project strong cuts from the tightest model onto the lighter models. We conclude this work with extensive numerical experiments to assess the quality of the mixed-integer linear formulations, as well as the performance of the cutting plane algorithms and the impact of the preprocessing on computation times
Benders decomposition for Network Design Covering Problems
We consider two covering variants of the network design problem. We are given a set of origin/destination (O/D) pairs and each such O/D pair is covered if there exists a path in the network from the origin to the destination whose length is not larger than a given threshold. In the first problem, called the maximal covering network design problem, one must determine a network that maximizes the total demand of the covered O/D pairs subject to a budget constraint on the design costs of the network. In the second problem, called the partial covering network design problem, the design cost is minimized while a lower bound is set on the total demand covered. After presenting formulations, we develop a Benders decomposition approach to solve the problems. Further, we consider two different stabilization methods to determine the Benders cuts as well as the addition of cut-set inequalities to the master problem. Computational experiments show the efficiency of these different aspects
An extended version of the Ordered Median Tree Location Problem including appendices and detailed computational results
In this paper, we propose the Ordered Median Tree Location Problem (OMT). The
OMT is a single-allocation facility location problem where p facilities must be
placed on a network connected by a non-directed tree. The objective is to
minimize the sum of the ordered weighted averaged allocation costs plus the sum
of the costs of connecting the facilities in the tree. We present different
MILP formulations for the OMT based on properties of the minimum spanning tree
problem and the ordered median optimization. Given that ordered median hub
location problems are rather difficult to solve we have improved the OMT
solution performance by introducing covering variables in a valid reformulation
plus developing two pre-processing phases to reduce the size of this
formulations. In addition, we propose a Benders decomposition algorithm to
approach the OMT. We establish an empirical comparison between these new
formulations and we also provide enhancements that together with a proper
formulation allow to solve medium size instances on general random graphs.Comment: 112 pages, 4 figures, extended version of 'The Ordered Median
Location Problem
“Facet" Separation with One Linear Program
Given polyhedron P and and a point x*, the separation problem for polyhedra asks to certify that x* ∈ P and if not, to determine an inequality that is satisfied by P and violated by x*. This problem is repeatedly solved in cutting plane methods for Integer Programming and the quality of the violated inequality is an essential feature in the performance of such methods. In the paper we address the problem of finding efficiently an inequality that is violated by x* and either defines an improper face or a facet of P. We provide some evidence that our method works on structured and unstructured problems