A New Algebraic Geometry Algorithm for Integer Programming

1996

Conventional methods for solving integer programming (IP) are based on heuristic searching algorithms. Recently, the tools of commutative algebra and algebraic geometry have bought new insights to integer programming via the theory of Gr obner bases. The key idea is to encode IP problems into a special ideal associated with the constraint matrix A and the cost (object) function C. An important property of the ideal is that its Gr obner bases correspond directly to the test sets of the IP problem. Using a proper test set, the optimal value of the cost function can be computed by constructing a monotonic path from the initial non-optimal solution of the problem to the optimal solution. Thus, IP can be solved without using intensive heuristic search. This approach is particularly interesting from the point of view of parallelism due to the inherent parallelism of the Buchberger algorithm that can be used to compute the Gr obner bases. This paper presents a parallel geometric Buchberger algorithm as an e cient parallel IP solver. A preliminary implementation of this algorithm has been carried out on a Fujitsu AP1000+.

Journal of Symbolic Computation, 2010

Multiobjective discrete programming is a well-known family of optimization problems with a large spectrum of applications. The linear case has been tackled by many authors during the past few years. However, the polynomial case has not been studied in detail due to its theoretical and computational difficulties. This paper presents an algebraic approach for solving these problems. We propose a methodology based on transforming the polynomial optimization problem to the problem of solving one or more systems of polynomial equations and we use certain Gröbner bases to solve these systems. Different transformations give different methodologies that are theoretically stated and compared by some computational tests via the algorithms that they induce.

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.

Mathematics of Operations Research, 1983

It is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable. The proof depends on methods from geometry of numbers.

Handbooks in Operations Research and Management Science, 2005

We review and describe several results regarding integer programming problems in fixed dimension. First, we describe various lattice basis reduction algorithms that are used as auxiliary algorithms when solving integer feasibility and optimization problems. Next, we review three algorithms for solving the integer feasibility problem. These algorithms are based on the idea of branching on lattice hyperplanes, and their running time is polynomial in fixed dimension. We also briefly describe an algorithm, based on a different principle, to count integer points in an integer polytope. We then turn the attention to integer optimization. Again, we describe three algorithms: binary search, a linear algorithm for a fixed number of constraints, and a randomized algorithm for a varying number of constraints. The topic of the next part of our chapter is how to use lattice basis reduction in problem reformulation. Finally, we review cutting plane results when the dimension is fixed.

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.

Lecture Notes in Computer Science, 2004

This paper presents three kinds of algebraic-analytic algorithms for solving integer and mixed integer programming problems. We report both theoretical and experimental results. We use the generating function techniques introduced by A. Barvinok.

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 ...

Mathematical Programming, 1982

Let A be a non-negative matrix with integer entries and no zero column. The integer round-up property holds for A if for every integral vector w the optimum objective value of the generalized covering problem min{1 y: yA > w, y > 0 integer} is obtained by rounding up to the nearest integer the optimum objective value of the corresponding linear program. A polynomial time algorithm is presented that does the following: given any generalized covering problem with constraint matrix A and right hand side vector w, the algorithm either finds an optimum solution vector for the covering problem or else it reveals that matrix A does not have the integer round-up property.

Journal of Optimization Theory and Applications, 1976

A computational comparison of several methods for dealing with polynomial geometric programs is presented. Specifically, we compare the complementary programs of Avriel and Williams (Ref. 1) with the reversed programs and the harmonic programs of Duffin and Peterson (Refs. 2, 3). These methods are used to generate a sequence of posynomial geometric programs which are solved using a dual algorithm.