On optimizing over lift-and-project closures

The lift-and-project closure is the relaxation obtained by computing all lift-and-project cuts from the initial formulation of a mixed integer linear program or equivalently by computing all mixed integer Gomory cuts read from all tableau's corresponding to feasible and infeasible bases. In this paper, we present an algorithm for approximating the value of the lift-and-project closure. The originality of our method is that it is based on a very simple cut generation linear programming problem which is obtained from the original linear relaxation by simply modifying the bounds on the variables and constraints. This separation LP can also be seen as the dual of the cut generation LP used in disjunctive programming procedures with a particular normalization. We study some properties of this separation LP in particular relating it to the equivalence between lift-and-project cuts and Gomory cuts shown by Balas and Perregaard. Finally, we present some computational experiments and comparisons with recent related works

Disjunctive Inequalities: Applications and Extensions

A general optimization problem can be expressed in the form min{cx: x ∈ S}, (1) where x ∈ R n is the vector of decision variables, c ∈ R n is a linear objective function and S ⊂ R n is the set of feasible solutions of (1). Because S is generall

Polyhedral Cones of Magic Cubes and Squares

Using computational algebraic geometry techniques and Hilbert bases of polyhedral cones we derive explicit formulas and generating functions for the number of magic squares and magic cubes.Comment: 14 page

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

Small Chvatal rank

We propose a variant of the Chvatal-Gomory procedure that will produce a sufficient set of facet normals for the integer hulls of all polyhedra {xx : Ax <= b} as b varies. The number of steps needed is called the small Chvatal rank (SCR) of A. We characterize matrices for which SCR is zero via the notion of supernormality which generalizes unimodularity. SCR is studied in the context of the stable set problem in a graph, and we show that many of the well-known facet normals of the stable set polytope appear in at most two rounds of our procedure. Our results reveal a uniform hypercyclic structure behind the normals of many complicated facet inequalities in the literature for the stable set polytope. Lower bounds for SCR are derived both in general and for polytopes in the unit cube.Comment: 24 pages, 3 figures, v3. Major revision: additional author, new application to stable-set polytopes, reorganization of sections. Accepted for publication in Mathematical Programmin

On the integration of Dantzig-Wolfe and Fenchel decompositions via directional normalizations

The strengthening of linear relaxations and bounds of mixed integer linear programs has been an active research topic for decades. Enumeration-based methods for integer programming like linear programming-based branch-and-bound exploit strong dual bounds to fathom unpromising regions of the feasible space. In this paper, we consider the strengthening of linear programs via a composite of Dantzig-Wolfe and Fenchel decompositions. We provide geometric interpretations of these two classical methods. Motivated by these geometric interpretations, we introduce a novel approach for solving Fenchel sub-problems and introduce a novel decomposition combining Dantzig-Wolfe and Fenchel decompositions in an original manner. We carry out an extensive computational campaign assessing the performance of the novel decomposition on the unsplittable flow problem. Very promising results are obtained when the new approach is compared to classical decomposition methods

Computational Experiments with Cross and Crooked Cross Cuts

In this paper, we study whether cuts obtained from two simplex tableau rows at a time can strengthen the bounds obtained by Gomory mixed-integer (GMI) cuts based on single tableau rows. We also study whether cross and crooked cross cuts, which generalize split cuts, can be separated in an effective manner for practical mixed-integer programs (MIPs) and can yield a nontrivial improvement over the bounds obtained by split cuts. We give positive answers to both these questions for MIPLIB 3.0 problems. Cross cuts are a special case of the t-branch split cuts studied by Li and Richard [Li Y, Richard J-PP (2008) Cook, Kannan and Schrijvers's example revisited. Discrete Optim. 5:724–734]. Split cuts are 1-branch split cuts, and cross cuts are 2-branch split cuts. Crooked cross cuts were introduced by Dash, Günlük, and Lodi [Dash S, Günlük O, Lodi A (2010) MIR closures of polyhedral sets. Math Programming 121:33–60] and were shown to dominate cross cuts by Dash, Günlük, and Molinaro [Dash S, Günlük O, Molinaro M (2012b) On the relative strength of different generalizations of split cuts. IBM Technical Report RC25326, IBM, Yorktown Heights, NY].United States. Office of Naval Research (Grant N000141110724

Algorithms for Stochastic Integer Programs Using Fenchel Cutting Planes

This dissertation develops theory and methodology based on Fenchel cutting planes for solving stochastic integer programs (SIPs) with binary or general integer variables in the second-stage. The methodology is applied to auto-carrier loading problem under uncertainty. The motivation is that many applications can be modeled as SIPs, but this class of problems is hard to solve. In this dissertation, the underlying parameter distributions are assumed to be discrete so that the original problem can be formulated as a deterministic equivalent mixed-integer program. The developed methods are evaluated based on computational experiments using both real and randomly generated instances from the literature. We begin with studying a methodology using Fenchel cutting planes for SIPs with binary variables and implement an algorithm to improve runtime performance. We then introduce the stochastic auto-carrier loading problem where we present a mathematical model for tactical decision making regarding the number and types of auto-carriers needed based on the uncertainty of availability of vehicles. This involves the auto-carrier loading problem for which actual dimensions of the vehicles, regulations on total height of the auto-carriers and maximum weight of the axles, and safety requirements are considered. The problem is modeled as a two-stage SIP, and computational experiments are performed using test instances based on real data. Next, we develop theory and a methodology for Fenchel cutting planes for mixed integer programs with special structure. Integer programs have to be solved to generate a Fenchel cutting plane and this poses a challenge. Therefore, we propose a new methodology for constructing a reduced set of integer points so that the generation of Fenchel cutting planes is computationally favorable. We then present the computational results based on randomly generated instances from the literature and discuss the limitations of the methodology. We finally extend the methodology to SIPs with general integer variables in the second-stage with special structure, and study different normalizations for Fenchel cut generation and report their computational performance