Theory of linear and integer programming

  title={Theory of linear and integer programming},
  author={Alexander Schrijver},
  booktitle={Wiley-Interscience series in discrete mathematics and optimization},
  • A. Schrijver
  • Published in
    Wiley-Interscience series in…
    1 December 1986
  • Mathematics, Computer Science
Introduction and Preliminaries. Problems, Algorithms, and Complexity. LINEAR ALGEBRA. Linear Algebra and Complexity. LATTICES AND LINEAR DIOPHANTINE EQUATIONS. Theory of Lattices and Linear Diophantine Equations. Algorithms for Linear Diophantine Equations. Diophantine Approximation and Basis Reduction. POLYHEDRA, LINEAR INEQUALITIES, AND LINEAR PROGRAMMING. Fundamental Concepts and Results on Polyhedra, Linear Inequalities, and Linear Programming. The Structure of Polyhedra. Polarity, and… 

Topics from this paper

On linear programming and matrix scaling over the algebraic numbers
Abstract Adler and Beling considered the linear programming problem over the real algebraic numbers. They obtained the necessary bounds in terms of a notion of size and dimension needed to justify
Polyhedral and Semidefinite Programming Methods in Combinatorial Optimization
Since the early 1960s, polyhedral methods have played a central role in both the theory and practice of combinatorial optimization. Since the early 1990s, a new technique, semidefinite programming,
New geometric techniques for linear programming and graph partitioning
The first randomized polynomial-time simplex algorithm for linear programming is presented, answering a question that has been open for over fifty years and an O(g/n) bound on the second smallest eigenvalue of the Laplacian of such graphs is proved and shown to be tight, thereby resolving a conjecture of Spielman and Teng.
Linear Programming Stories
The history of polyhedra, linear inequalities, and linear programming has many diverse origins. Polyhedra have been around since the beginning of mathematics in ancient times. It appears that Fourier
Integer programming, lattice algorithms, and deterministic volume estimation
The main subject of this thesis is the development of new geometric tools and techniques for solving classic problems within the geometry of numbers and convex geometry. At a high level, the problems
Subdeterminants and Concave Integer Quadratic Programming
An algorithm is given that finds an epsilon-approximate solution for the NP-hard problem of minimizing a separable concave quadratic function over the integral points in a polyhedron by solving a number of integer linear programs whose constraint matrices have subdeterminants bounded by D in absolute value.
Polyhedral Omega: a New Algorithm for Solving Linear Diophantine Systems
This work combines methods from partition analysis with methods from polyhedral geometry, and provides an illustrated geometric interpretation of partition analysis, with the aim of making ideas from both areas accessible to readers from a wide range of backgrounds.
Convex Discrete Optimization
  • S. Onn
  • Mathematics, Computer Science
    Encyclopedia of Optimization
  • 2009
Using a combination of algebraic and geometric tools, this work provides polynomial time algorithms for solving broad classes of convex combinatorial optimization problems and convex integer programming problems in variable dimension.
Polyhedral techniques in combinatorial optimization II: applications and computations
The polyhedral approach is one of the most powerful techniques available for solving hard combinatorial optimization problems. The main idea behind the technique is to consider the linear relaxation
Computational algebraic analyses for unimodular or Lawrence-type integer programs
There have been a number of studies of the application of computational algebraic approaches to solve integer programming problems, using Gr ̈ obner bases or standard pairs. By connecting these