Gomory Cuts Research Articles

Chamfer Distance on the Four-dimensional Face-centred Cubic Grid

Chamfer distances are step-based distances on various grids. The fourdimensional FCC grid is the extension of the usual face-centred cubic grid into four dimensions. There are two types of steps, and thus, two weights are used. Operational research, namely linear programming and Gomory cut are applied to describe optimal paths and hence, their weighted lengths, the chamfer distance.

On Chamfer Distances on the Square and Body-Centered Cubic Grids: An Operational Research Approach

Linear programming is used to solve optimization problems. Thus, finding a shortest path in a grid is a good target to apply linear programming. In this paper, specific bipartite grids, the square and the body-centered cubic grids are studied. The former is represented as a “diagonal square grid” having points with pairs of either even or pairs of odd coordinates (highlighting the bipartite feature). Therefore, a straightforward generalization of the representation describes the body-centered cubic grid in 3D. We use chamfer paths and chamfer distances in these grids; therefore, weights for the steps between the closest neighbors and steps between the closest same type points are fixed, and depending on the weights, various paths could be the shortest one. The vectors of the various neighbors form a basis if they are independent, and their number is the same as the dimension of the space studied. Depending on the relation of the weights, various bases could give the optimal solution and various steps are used in the shortest paths. This operational research approach determines the optimal paths as basic feasible solutions of a linear programming problem. A directed graph is given containing the feasible bases as nodes and arcs with conditions on the used weights such that the simplex method may step from one feasible basis to another one. Thus, the optimal bases can be determined, and they are summarized in two theorems. If the optimal solution is not integer, then the Gomory cut is applied and the integer optimal solution is reached after only one Gomory iteration. Chamfer distances are frequently used in image processing and analysis as well as graphics-related subjects. The body-centered cubic grid, which is well-known in solid state physics, material science, and crystallography, has various applications in imaging and graphics since less samples are needed to represent the signal in the same quality than on the cubic grid. Moreover, the body-centered cubic grid has also a topological advantage over the cubic grid, namely, the neighbor Voronoi cells always share a full face.

How to find the convex hull of all integer points in a polyhedron?

We propose a cut-based algorithm for finding all vertices and all facets of the convex hull of all integer points of a polyhedron defined by a system of linear inequalities. Our algorithm, DDMCuts, is based on the Gomory cuts and the dynamic version of the double description method. We describe the computer implementation of the algorithm and present the results of computational experiments comparing our algorithm with a naive one and an algorithm implemented in Normaliz.

Chvátal–Gomory cuts for the Steiner tree problem

The Steiner tree problem asks for a minimum weight subtree of an undirected graph spanning distinct terminal nodes. As one of the most fundamental network design problems, it has received much attention from both the mathematical programming and the algorithmic community and indeed, many integer linear programming (ILP) formulations of varying strength are known. The practical usefulness of the formulations can be enhanced with additional valid inequalities that reduce the gap between the efficiently computable continuous relaxation and the computationally difficult integer linear program. We aim to better understand the structure of these inequalities. In particular, we want to know how the inequalities can be obtained through Chvátal–Gomory cuts; a general procedure that is implemented in state-of-the-art ILP solver software. We start with the most basic ILP formulation for the Steiner tree problem and analyze how two known classes of strengthening inequalities – partition inequalities and odd-hole inequalities – can be derived systematically as Chvátal–Gomory cuts and how these inequalities themselves can be used to derive Chvátal–Gomory cuts. Along the way, we prove that the Chvátal rank of k-partition and k-odd-hole inequalities is at most k−2 and k−1, respectively. In particular, this proves that k iterations of the Chvátal–Gomory cut procedure suffice to obtain a given k+2-partition or a given k+1-odd-hole inequality.

On matroid parity and matching polytopes

The matroid parity (MP) problem is a powerful (and NP-hard) extension of the matching problem. Whereas matching polytopes are well understood, little is known about MP polytopes. We prove that, when the matroid is laminar, the MP polytope is affinely congruent to a perfect b-matching polytope. From this we deduce that, even when the matroid is not laminar, every Chvátal–Gomory cut for the MP polytope can be derived as a {0,12}-cut from a laminar family of rank constraints. We also prove a negative result concerned with the integrality gap of two linear relaxations of the MP problem.

High Degree Sum of Squares Proofs, Bienstock--Zuckerberg Hierarchy, and Chvátal--Gomory Cuts

Chvátal--Gomory cuts (CG-cuts) and the Bienstock--Zuckerberg hierarchy capture useful linear programs that the standard bounded degree sum-of-squares (SoS) hierarchy fails to capture. In this paper we present a novel polynomial time SoS hierarchy for 0/1 problems with a custom subspace of high degree polynomials (not the standard subspace of low degree polynomials). We show that the new SoS hierarchy recovers the Bienstock--Zuckerberg hierarchy. Our result implies a linear program that reproduces the Bienstock--Zuckerberg hierarchy as a polynomial-sized, efficiently constructible extended formulation that satisfies all constant pitch inequalities. The construction is also very simple, and it is fully defined by giving the supporting polynomials. Moreover, for a class of polytopes (e.g., set cover and packing problems), the resulting SoS hierarchy optimizes in polynomial time over the polytope resulting from any constant rounds of CG-cuts, up to an arbitrarily small error in the solution value. Arguably, this is the first example where different basis functions can be useful in asymmetric situations to obtain a hierarchy of relaxations.

Aggregation-based cutting-planes for packing and covering integer programs

In this paper, we study the strength of Chvatal---Gomory (CG) cuts and more generally aggregation cuts for packing and covering integer programs (IPs). Aggregation cuts are obtained as follows: given an IP formulation, we first generate a single implied inequality using aggregation of the original constraints, then obtain the integer hull of the set defined by this single inequality with variable bounds, and finally use the inequalities describing the integer hull as cutting-planes. Our first main result is to show that for packing and covering IPs, the CG and aggregation closures can be 2-approximated by simply generating the respective closures for each of the original formulation constraints, without using any aggregations. On the other hand, we use computational experiments to show that aggregation cuts can be arbitrarily stronger than cuts from individual constraints for general IPs. The proof of the above stated results for the case of covering IPs with bounds require the development of some new structural results, which may be of independent interest. Finally, we examine the strength of cuts based on k different aggregation inequalities simultaneously, the so-called multi-row cuts, and show that every packing or covering IP with a large integrality gap also has a large k-aggregation closure rank. In particular, this rank is always at least of the order of the logarithm of the integrality gap.

Another pedagogy for pure-integer Gomory

We present pure-integer Gomory cuts in a way so that they are derived with respect to a "dual form" pure-integer optimization problem and applied on the standard-form primal side as columns, using the primal simplex algorithm. The input integer problem is not in standard form, and so the cuts are derived a bit differently. In this manner, we obtain a finitely-terminating version of pure-integer Gomory cuts that employs the primal rather than the dual simplex algorithm.

Integer programming as projection

We generalise polyhedral projection (Fourier–Motzkin elimination) to integer programming (IP) and derive from this an alternative perspective on IP that parallels the classical theory. We first observe that projection of an IP yields an IP augmented with linear congruence relations and finite-domain variables, which we term a generalised IP. The projection algorithm can be converted to a branch-and-bound algorithm for generalised IP in which the search tree has bounded depth (as opposed to conventional branching, in which there is no bound). It also leads to valid inequalities that are analogous to Chvátal–Gomory cuts but are derived from congruences rather than rounding, and whose rank is bounded by the number of variables. Finally, projection provides an alternative approach to IP duality. It yields a value function that consists of nested roundings as in the classical case, but in which ordinary rounding is replaced by rounding to the nearest multiple of an appropriate modulus, and the depth of nesting is again bounded by the number of variables. For large perturbations of the right-hand sides, the value function is shift periodic and can be interpreted economically as yielding “average” shadow prices.

Single-commodity robust network design with finite and Hose demand sets

We study a single-commodity robust network design problem (sRND) defined on an undirected graph. Our goal is to determine minimum cost capacities such that any traffic demand from a given uncertainty set can be satisfied by a feasible single-commodity flow. We consider two ways of representing the uncertainty set, either as a finite list of scenarios or as a polytope. We propose a branch-and-cut algorithm to derive optimal solutions to sRND, built on a capacity-based integer linear programming formulation. It is strengthened with valid inequalities derived as $$\{0,\frac{1}{2}\}$${0,12}-Chvatal---Gomory cuts. Since the formulation contains exponentially many constraints, we provide practical separation algorithms. Extensive computational experiments show that our approach is effective, in comparison to existing approaches from the literature as well as to solving a flow based formulation by a general purpose solver.

Cut-Generating Functions and S-Free Sets

We consider the separation problem for sets X that are pre-images of a given set S by a linear mapping. Classical examples occur in integer programming, as well as in other optimization problems such as complementarity. One would like to generate valid inequalities that cut off some point not lying in X, without reference to the linear mapping. To this aim, we introduce a concept: cut-generating functions (CGF) and we develop a formal theory for them, largely based on convex analysis. They are intimately related to S-free sets and we study this relation, disclosing several definitions for minimal CGF's and maximal S-free sets. Our work unifies and puts into perspective a number of existing works on S-free sets; in particular, we show how CGF's recover the celebrated Gomory cuts.

Fuzzy Bi-Level Multi-Objective Fractional Integer Programming

We present an algorithm to solve a bi-level multi-objective fractional integer programming problem involving fuzzy numbers in the right-hand side of the constraints. The suggested algorithm combine the method of Taylor series together with the Kuhn Tucker conditions to solve fuzzy bi-level multi-objective fractional integer programming problem (FBLMOFIPP) then Gomory cuts are added till the integer solution is obtained. An illustrative example is discussed to demonstrate the correctness of the proposed solution method.

A New Procedure for Solving Integer Linear Programming Problems

This paper presents a new procedure for solving the integer linear programming problem when the objective function is a linear function and the set of constraints is in the form of linear inequality constraints. The proposed procedure is based on the conjugate gradient projection method together with the use of the spirit of Gomory cut. The main idea behind our method is to move through the feasible region through a sequence of points in the direction that improves the objective function. Since methods based on vertex information may have difficulties as the problem size increases, therefore, the present method can be considered as an interior point method, which had been proved to be less sensitive to problem size. A simple production example is given to clarify the theory of this new procedure.

Iterated Chvátal--Gomory Cuts and the Geometry of Numbers

Chvátal--Gomory cutting planes (CG-cuts for short) are a fundamental tool in integer programming. Given any single CG-cut, one can derive an entire family of CG-cuts, by “iterating" its multiplier vector modulo one. This leads naturally to two questions: first, which iterates correspond to the strongest cuts, and, second, can we find such strong cuts efficiently? We answer the first question empirically, by showing that one specific approach for selecting the iterate tends to perform much better than several others. The approach essentially consists of solving a nonlinear optimization problem over a special lattice associated with the CG-cut. We then provide a partial answer to the second question, by presenting a polynomial-time algorithm that yields an iterate that is strong in a certain well-defined sense. The algorithm is based on results from the algorithmic geometry of numbers.

Finitely Convergent Decomposition Algorithms for Two-Stage Stochastic Pure Integer Programs

We study a class of two-stage stochastic integer programs with general integer variables in both stages and finitely many realizations of the uncertain parameters. Based on Benders' method, we propose a decomposition algorithm that utilizes Gomory cuts in both stages. The Gomory cuts for the second-stage scenario subproblems are parameterized by the first-stage decision variables, i.e., they are valid for any feasible first-stage solutions. In addition, we propose an alternative implementation that incorporates Benders' decomposition into a branch-and-cut process in the first stage. We prove the finite convergence of the proposed algorithms. We also report our preliminary computations with a rudimentary implementation of our algorithms to illustrate their effectiveness.

Solving Integer and Mixed Integer Linear Problems with ABS Method

Solving mixed integer linear programming (MILP) problems is a difficult task due to the parallel use of both integer and non-integer values. One of the most widely used solution is to solve the problem in the real space and they apply additional iteration steps (so-called cutting-plane algorithms or Gomory's cuts) to narrow down the solution to the optimal integer solution. The ABS class of algorithms is a generalized class of algorithms which, with appropriate selection of parameters, is suitable for the solution of both integer and non-integer linear problems. Here we provide for the first time a complete ABS-based algorithm for MILP problems by adaptation of the ABS approach to Gomory's cutting- plane algorithm. We also provide a numerical example demonstrating the working principle of our algorithm.

On the safety of Gomory cut generators

Gomory mixed-integer cuts are one of the key components in Branch-and-Cut solvers for mixed-integer linear programs. The textbook formula for generating these cuts is not used directly in open-source and commercial software that work in finite precision: Additional steps are performed to avoid the generation of invalid cuts due to the limited numerical precision of the computations. This paper studies the impact of some of these steps on the safety of Gomory mixed-integer cut generators. As the generation of invalid cuts is a relatively rare event, the experimental design for this study is particularly important. We propose an experimental setup that allows statistically significant comparisons of generators. We also propose a parameter optimization algorithm and use it to find a Gomory mixed-integer cut generator that is as safe as a benchmark cut generator from a commercial solver even though it generates many more cuts.

Lifting Gomory cuts with bounded variables

Recently, Balas and Qualizza introduced a new cut for mixed 0,1 programs, called lopsided cut. Here we present a family of cuts that comprises the Gomory mixed integer cut at one extreme and the lopsided cut at the other.

Decomposition algorithms with parametric Gomory cuts for two-stage stochastic integer programs

We consider a class of two-stage stochastic integer programs with binary variables in the rst stage and general integer variables in the second stage. We develop decomposition algorithms akin to the L-shaped or Benders’ methods by utilizing Gomory cuts to obtain iteratively tighter approximations of the second-stage integer programs. We show that the proposed methodology is exible in that it allows several modes of implementation, all of which lead to nitely convergent algorithms. We illustrate our algorithms using examples from the literature. We report computational results using the stochastic server location problem instances which suggest that our decomposition-based approach scales better with increases in the number of scenarios than a state-of-the art solver which was used to solve the deterministic equivalent formulation.

Wiley encyclopedia of operations research and management science

Use of Lagrange Interpolating Polynomials in RLT. Systems in Series. Parallel Configurations. Spanning trees. Recycling. Split Cuts. Retrial Queues. Horse Racing. MACBETH (Measuring Attractiveness by a Categorical Based Evaluation Technique). Dual simplex. History of CP. Procurement Contracts. MRP. Just in Time. Push and Pull Production. MULTIVARIATE INPUT MODELING. Operational Research Society. The M/G/1 Queue. Drama Theory. Hyper-heuristics. Antithetic Variates. Music. Edgeworth market games. Operational Risk. Queueing Notation. Environmental impact. Why variance is not risk. Conjugate gradient methods. Birth-and-Death Processes. Surgery Planning and Scheduling. Large-scale linear models. Standby Systems. Billiards/Pool. AXIOMATIC MEASURES OF RISK AND RISK-VALUE MODELS. Age Replacement Policies. PASTA and Related Results. Collaborative Procurement. LP projection algorithms. R&D risk management. Bioterrorism. Vendor-Managed Inventory. Gomory Cuts. Travel demand modeling. System Availability. Infinite horizon problems. Trust. Bayesian Network Classifiers. Integer Programming Duality. Queueing Disciplines. Stakeholder participation. Models and Basic Properties. Parametric LP analysis. Operations Research and Golfing. Ice Hockey. Clustering. Finite-Population Models. Batch Arrivals and Service. Optimal Monitoring Strategies. Basic Polyhedral Theory. Operations Research and Art. Managing portfolios of risks. Measures of risk equity. Branch and Bound Algorithms. Analytics in Retail. Variational Inequalities. Cross-Entropy Method. TSP heuristics. Capacity allocation. Control Variates. Cover Inequalities. Direct Search Methods. NLP Software. Bilinear optimization. Discrete-Time Martingales. Branch and Price. Column generation. IP Preprocessing. Optimal Reliability Allocation. Simplex-based LP solvers. Instance Formats. Clique generalizations. Availability in Stochastic Models. Evolutionary Algorithms. Basic CP theory: Search. Multimethodology. Branch and Cut. Memetic Algorithms. Urban mass transit. Coherent Systems. Subgradient optimization. Klimov's Model. Credit risk assessment. Branch-width and Tangles. Stochastic Hazard Process. Deterministic Global Optimization. Multistage (stochastic) games. Basis Reduction Methods. ACCIDENT PRECURSORS AND WARNING SYSTEMS MANAGEMENT: A BAYESIAN APPROACH TO MATHEMATICAL MODELS. Genetic Algorithms. Newsvendor models. Parallel Systems. Opportunities in Tennis. Job-shop scheduling. Simplex method and complexity. Geometric programming. Evacuation Planning. The Performance? Allocation Games. Scenario Generation. Robustness Analysis. Dynamic auctions. Neuroeconomics and game theory. Common Random Numbers. Selective Support Vector Machines. Semi-Markov Processes. Risk Averse Models. Sampling Methods. Domination problems. Simulation of Rare Events. Introduction to Tabu Search. Basic Interdiction Models. Introductory Concepts. Remanufacturing. PAUSE procedure. Wardrop equilibria. Mass customization. Ellipsoidal algorithms. Benders decomposition. Dynamic vehicle routing. Warranty Modeling. Design for Network Resiliency. Passenger rail transportation. Vehicle Routing Problem. Supply Chain Outsourcing. Lovasz-Schriver Reformulation. Hazard Rate Function. Lot-sizing. Differential games. Regenerative Processes. Guided Local Search. Preferences for repeated gambles. Percolation Theory. Simulated Annealing. Scoring Rules. Mixing Sets. Cricket. Prospect Theory. Combining Forecasts. Graph search techniques. Quasi-Newton methods. Random search algorithms. Supply chain coordination. Service Outsourcing. Markov Renewal Processes. K-out-of-n Systems.