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About 10,000 papers for “Integer programming”

Integer programming : dual all-integer integer programming (Gomory [2])

Salkin, Harvey M. · 1971

This work represents the sixth chapter of a forthcoming textbook “Integer Programming” to be written by Harvey M. Salkin. The cutting plane algorithm for the integer program presented in this chapter was developed by Ralph Gomory in 1960. Its similarity to the fractional method (Chapter 3) is principally due to the utilization of the (lexicographic) dual simplex method and the maintenance of (lexicographic) positive columns. The basic approach is, however, different from the fractional technique: there is no optimization, generating a constraint, reoptimization, etc. Rather, inequalities are generated at each iteration starting with the very first. Further, each of these constraints is used as the pivot row and is constructed so that it has integral coefficients and the pivot is -1. The initial tableau is assumed to be all integer and dual feasible. Hence, successive tableaux are also all integer and dual feasible. The primal solution proceeds towards feasibility, and since dual feasibility is maintained, optimality is reached when it is attained. Note that the method is a direct extension of the classical dual simplex algorithm. The essential difference is that the pivot row in the all-integer algorithm is generated at each iteration and ensures a -1 pivot. Since the technique employs the dual simplex method and maintains all integer tableaux, it is referred to as "dual all-integer." We list the steps.

Cited by 0Open access

Integer Programming

2021

An integer linear program is a program whose variables are restricted to integer values. Many optimization problems require a solution with integer values. Many constraints that seem nonlinear can be linearized because of integer variables. This chapter presents specialized methods for solving this type of program, including: the branch-and-bound method and the branch-and-cut method. The branch-and-bound method essentially consists of the following three key elements: a separation procedure, an evaluation procedure and a pathing procedure. The knapsack problem is one of the most common problems in binary programming. Lagrangian relaxation is applied to the coupling constraints by adding them to the objective function. Benders decomposition method is dedicated to mixed problems. It proceeds by solving a master problem containing only real variables and multiple subproblems with integer variables. To work with integer-valued and mixed programs, Matlab offers the intlinprog function.

Cited by 0

Integer programming.

Zhen Peng (539724), Jeff Linderoth (13767980), David A. Baum (13767983) · Figshare · 2022

Cited by 0Open access

Integer Programming

Michele Conforti, Gérard Cornuéjols, Giacomo Zambelli · Graduate texts in mathematics · 2014

Cited by 186

Integer Programming

Michael Carter, Camille C. Price, Ghaith Rabadi · Operations Research · 2018

Cited by 0

Integer Programming

Rajput, Rohini R. · e-Publications@Marquette (Marquette University) · 1987

Mathematical programming problems in which all variables can assume nonnegative integer values only are called pure integer programming problems. If in the absence of the integrality conditions the objective and constraint functions are linear, the resulting model is called an integer linear program. Such problems are hard to solve both in theory and in practice. One common approach is to solve the continuous version of the problem and then round the continuous optimum to the closest feasible integers. Here rounding implies approximation, so one cannot guarantee that the rounded solution will always satisfy the constraints, and the rounded solution cannot be feasible in linear programming problem since same basis can yield two different solutions

Cited by 0

Integer Programming

Ellis L. Johnson · Society for Industrial and Applied Mathematics eBooks · 1980

Integer Programming Cuts, Knapsacks, and a Cyclic Group Problem Finite Abelian Groups Gomory's Corner Polyhedra Blocking Polyhedra and Master Group Problems Araoz's Semigroup Problem Blockers and Polars for Master Semigroup Problems Subadditive and Minimal Valid Inequalities Subadditive Characterizations Duality.

Cited by 36

Integer Programming

José María Ponce‐Ortega, Rogelio Ochoa-Barragán, César Ramírez‐Márquez · 2024

Cited by 1

Integer programming

G. L. Nemhauser, Laurence A. Wolsey · RePEc: Research Papers in Economics · 1972

The principles of integer programming are directed toward finding solutions to problems from the fields of economic planning, engineering design, and combinatorial optimization. This highly respected and much-cited text, a standard of graduate-level courses since 1972, presents a comprehensive treatment of the first two decades of research on integer programming.

Cited by 3,383

Integer Programming

Egon Balas · Wiley StatsRef: Statistics Reference Online · 2015

Abstract Integer programming (IP) is optimization of a linear function subject to linear constraints in variables some or all of which are constrained to be integer. By far the most important special case of IP is 0–1 programming, in which the integer variables are restricted to 0 or 1. This is so because binary variables can be used to represent all kinds of situations not captured by linear or convex functions, like discontinuities, interruptions, implications, and other logical conditions that arise frequently in practice. The article outlines why and how this happens, discusses the scope and applicability of IP, and after introducing combinatorial optimization as the study of 0–1 programs defined on graphs, it reviews the relation of IP to statistics, in particular its applications to regression analysis, design of experiments and cluster analysis. Next, solution methods for IP's are outlined, the two principal ones being branch and bound (or implicit enumeration) and cutting planes (or convexification). The former one is relatively straightforward to implement and for 30 years was the only one used in practice, limiting the sphere of solvable IP's to no more than 30–40 integer variables. The latter one on the other hand generated a rich theory of cutting planes derived from several basic principles, but for 30 years could not be used in practice, mainly because of numerical problems. The article then outlines the revolution in the state of the art of IP that occurred roughly between 1990 and 2005. Namely, by embedding cutting planes into a branch‐and‐bound framework and using them in new ways, highly efficient algorithms were developed that can routinely solve instances with thousands of variables and constraints.

Cited by 0