# Primal cutting plane algorithms revisited

@article{Letchford2002PrimalCP, title={Primal cutting plane algorithms revisited}, author={Adam N. Letchford and Andrea Lodi}, journal={Mathematical Methods of Operations Research}, year={2002}, volume={56}, pages={67-81} }

Abstract.Dual fractional cutting plane algorithms, in which cutting planes are used to iteratively tighten a linear relaxation of an integer program, are well-known and form the basis of the highly successful branch-and-cut method. It is rather less well-known that various primal cutting plane algorithms were developed in the 1960s, for example by Young. In a primal algorithm, the main role of the cutting planes is to enable a feasible solution to the original problem to be improved. Research…

## 31 Citations

### Integral simplex using decomposition with primal cutting planes

- Computer ScienceMath. Program.
- 2017

It is shown that primal cuts, that is, cutting planes that are tight at the current feasible integer solution, can be used to improve the performance of the algorithm and further that such cutting planes are enough to solve each augmentation problem.

### Primal Cutting Plane Methods for the Traveling Salesman Problem

- Business
- 2017

Most serious attempts at solving the traveling salesman problem (TSP) are based on the dual fractional cutting plane approach, which moves from one lower bound to the next. This thesis describes…

### Integral Simplex Using Decomposition with Primal Cuts

- Computer ScienceSEA
- 2014

It is shown that MRA canonically induces a decomposition of the augmentation problem and deepens the understanding of ISUD, and characterize cuts that adapt to this decomposition and relate them to primal cuts.

### Linear-Programming-Based Lifting and Its Application to Primal Cutting-Plane Algorithms

- MathematicsINFORMS J. Comput.
- 2009

This lifting procedure can be applied to improve Gomory's fractional cut, which is central to Glover's primal cutting-plane algorithm, and it is shown that the resulting algorithm is finitely convergent.

### A Primal Branch-and-Cut Algorithm for the Degree-Constrained Minimum Spanning Tree Problem

- Computer ScienceWEA
- 2007

A primal branch-and-cut algorithm that solves instances of the degree-constrained minimum spanning tree problem to optimality and turns out to be competitive with other methods known in the literature.

### An Augment-and-Branch-and-Cut Framework for Mixed 0-1 Programming

- Computer ScienceCombinatorial Optimization
- 2001

A possible implementation of a finite ABC algorithm that differs from standard branch-and-cut in several important ways, including the terms separation, branching, and fathoming take on new meanings in the primal context.

### Primal separation algorithms

- Mathematics4OR
- 2003

The complexity of primal separation for several well-known classes of inequalities for various important combinatorial optimization problems, including the knapsack, stable set and travelling salesman problems are examined.

### Integral Column Generation for Set Partitioning Problems with Side Constraints

- Computer ScienceINFORMS Journal on Computing
- 2022

A new integral column generation algorithm that can solve efficiently large-scale set partitioning problems with side constraints is developed and the latter alter the quasi-integrality property needed for primal integral algorithms.

### Integral simplex using double decomposition for set partitioning problems

- Computer ScienceComput. Oper. Res.
- 2019

## References

SHOWING 1-10 OF 39 REFERENCES

### A lift-and-project cutting plane algorithm for mixed 0–1 programs

- Computer ScienceMath. Program.
- 1993

We propose a cutting plane algorithm for mixed 0–1 programs based on a family of polyhedra which strengthen the usual LP relaxation. We show how to generate a facet of a polyhedron in this family…

### A Branch-and-Cut Algorithm for the Resolution of Large-Scale Symmetric Traveling Salesman Problems

- Computer ScienceSIAM Rev.
- 1991

An algorithm is described for solving large-scale instances of the Symmetric Traveling Salesman Problem (STSP) to optimality. The core of the algorithm is a “polyhedral” cutting-plane procedure that…

### The integral basis method for integer programming

- Computer ScienceMath. Methods Oper. Res.
- 2001

An exact algorithm for solving integer programs, neither using cutting planes nor enumeration techniques, that relies on iteratively substituting one column by columns that correspond to irreducible solutions of certain linear diophantine inequalities.

### Primal separation algorithms

- Mathematics4OR
- 2003

The complexity of primal separation for several well-known classes of inequalities for various important combinatorial optimization problems, including the knapsack, stable set and travelling salesman problems are examined.

### A Simplified Primal (All-Integer) Integer Programming Algorithm

- Computer ScienceOper. Res.
- 1968

The simplified primal algorithm makes these major amendments to the simplex method: a special row, indexed by L, is adjoined to the tableau and is periodically revised by a well-defined procedure.

### Branch and cut algorithms

- Mathematics
- 1996

This paper proposes a basic solution strategy to divide a region into a number of smaller regions and optimize the objective function over each smaller region individually, to ensure that an optimal solution to (1.1) is contained in at least one of the smaller regions that were generated.

### A Bounding Minimization Problem for Primal Integer Programming

- Computer Science, MathematicsOper. Res.
- 1974

An algorithm for obtaining an upper bound on the value of the objective function based on the best bound obtainable from dual solutions to a class of related linear programs is described.

### A primal (all-integer) integer programming algorithm

- Computer Science
- 1965

The algorithm is most closely related to three existing procedures: the simplex method of G. B. Dantzig for linear programming problems, the Gumory all-integer integer programming algurithm, and the…

### Solving Multiple Knapsack Problems by Cutting Planes

- Computer ScienceSIAM J. Optim.
- 1996

The inequalities that are described here serve as the theoretical basis for a cutting plane algorithm that is applied to practical problem instances arising in the design of main frame computers, in the layout of electronic circuits, and in sugar cane alcohol production.

### AN ALGORITHM FOR THE MIXED INTEGER PROBLEM

- Mathematics
- 1960

Abstract : An algorithm is given for the numerical solution of the 'mixed integer' linear programming problem, the problem of maximizing a linear form in finitely many variables constrained both by…