A computational study of integer programming algorithms based on Barvinok's rational functions (original) (raw)
AN ALGORITHM FOR SOLVING INTEGER LINEAR PROGRAMMING PROBLEMS
The paper describes a method to solve an ILP by describing whether an approximated integer solution to the RLP is an optimal solution to the ILP. If the approximated solution fails to satisfy the optimality condition, then a search will be conducted on the optimal hyperplane to obtain an optimal integer solution using a modified form of Branch and Bound Algorithm.
A Computational Status Update for Exact Rational Mixed Integer Programming
2021
The last milestone achievement for the roundoff-error-free solution of general mixed integer programs over the rational numbers was a hybrid-precision branch-and-bound algorithm published by Cook, Koch, Steffy, and Wolter in 2013. We describe a substantial revision and extension of this framework that integrates symbolic presolving, features an exact repair step for solutions from primal heuristics, employs a faster rational LP solver based on LP iterative refinement, and is able to produce independently verifiable certificates of optimality. We study the significantly improved performance and give insights into the computational behavior of the new algorithmic components. On the MIPLIB 2017 benchmark set, we observe an average speedup of 6.6x over the original framework and 2.8 times as many instances solved within a time limit of two hours.
A HEURISTIC FOR GENERAL INTEGER PROGRAMMING
Decision Sciences, 1974
When constructing linear programming models one is frequently faced with the restriction that all variables must be integer valued. Integer restrictions are generally required because of the indivisible nature of the items t o be optimized e.g., in hiring it is meaningless to hire fractional people.
Contemporary Approaches to the Solution of the Integer Programming Problem
2003
The purpose of this thesis is to provide analysis of the modem development of the methods for solution to the integer linear programming problem. The thesis simultaneously discusses some main approaches that lead to the development of the algorithms for the solution to the integer linear programming problem. Chapter 1 introduces the Generalized Linear Programming Problem alongside with the properties of a solution to the Linear Programming Problem. The simplex procedure presented to solve the Linear Programming Problem by adding slack variables along with the artificial-basis technique. Chapter 2 refers to the primal-dual simplex procedure. The dual simplex algorithm reflects the dual simplex procedure. Chapter 3 discusses the mixed and alternative formulations of the integer programming problem. Chapter 4 considers the optimality conditions with the imposed relaxations to solve the Linear Programming Relaxation Problem. The methods of the Integer Programming are introduced for the ...
Non-standard approaches to integer programming
Discrete Applied Mathematics, 2002
In this survey we address three of the principal algebraic approaches to integer programming. After introducing lattices and basis reduction, we ÿrst survey their use in integer programming, presenting among others Lenstra's algorithm that is polynomial in ÿxed dimension, and the solution of diophanine equations using basis reduction. The second topic concerns augmentation algorithms and test sets, including the role played by Hilbert and Gr obner bases in the development of a primal approach to solve a family of problems for all right-hand sides. Thirdly we survey the group approach of Gomory, showing the importance of subadditivity in integer programming and the generation of valid inequalities, as well the relation to the parametric problem cited above of solving for all right-hand sides.
A New Algebraic Geometry Algorithm for Integer Programming
Management Science, 2000
W e propose a new algorithm for solving integer programming (IP) problems that is based on ideas from algebraic geometry. The method provides a natural generalization of the Farkas lemma for IP, leads to a way of performing sensitivity analysis, offers a systematic enumeration of all feasible solutions, and gives structural information of the feasible set of a given IP. We provide several examples that offer insights on the algorithm and its properties.
On The Approximation Of Real Rational Function Via Mixed Integer Linear Programming
This paper is introducing a new method for the approximation of real rational functions via mixed-integer linear programming. The formulation of the linear approximation problem is based on the minimization of a suitable minimax criterion, in combination with a branch and bound linear integer technique. The proposed algorithm can be used in many rational approximation problems, where some coecients of the rational function are required to take only integer values. The formulation of the problem ensures always the global solution. The proposed algorithm was extensively tested on a variety of problems. An analytical example is presented to illustrate the use and eectiveness of the algorithm. Ó
Arxiv preprint arXiv:0712.4295, 2007
This paper presents a new complexity result for solving multiobjective integer programming problems. We prove that encoding the entire set of nondominated solutions of the problem in a short sum of rational functions is polynomially doable, when the dimension of the decision space is fixed. This result extends a previous result presented in De Loera et al. (INFORMS J. Comput. 21(1):39–48, 2009) in that there the number of the objective functions is assumed to be fixed whereas ours allows this number to vary.