An integer linear programming approach for bilinear integer programming

We introduce a new Integer Linear Programming (ILP) approach for solving Integer Programming (IP) problems with bilinear objectives and linear constraints. The approach relies on a series of ILP approximations of the bilinear IP. We compare this approach with standard linearization techniques on random instances and a set of real-world product bundling problems.     

Parametric methods in integer linear programming

In contrast to methods of parametric linear programming which were developed soon after the invention of the simplex algorithm and are easily included as an extension of that method, techniques for parametric analysis on integer programs are not well known and require considerable effort to append them to an integer programming solution algorithm.The paper reviews some of the theory employed in parametric integer programming, then discusses algorithmic work in this area over the last 15 years when integer programs are solved by different methods. A summary of applications is included and the article concludes that parametric integer programming is a valuable tool of analysis awaiting further popularization.     

Sensitivity theorems in integer linear programming

We consider integer linear programming problems with a fixed coefficient matrix and varying objective function and right-hand-side vector. Among our results, we show that, for any optimal solution to a linear program max{wx: Axb}, the distance to the nearest optimal solution to the corresponding integer program is at most the dimension of the problem multiplied by the largest subdeterminant of the integral matrixA. Using this, we strengthen several integer programming proximity results of Blair and Jeroslow; Graver; and Wolsey. We also show that the Chvátal rank of a polyhedron {x: Axb} can be bounded above by a function of the matrixA, independent of the vectorb, a result which, as Blair observed, is equivalent to Blair and Jeroslow's theorem that each integer programming value function is a Gomory function.Supported by a grant from the Alexander von Humboldt Stiftung.Since September 1985: Department of Operations Research, Upson Hall, Cornell University, Ithaca, NY 14853, USA.Partially supported by the Sonderforschungbereich 21 (DFG), Institut für Ökonometrie und Operations Research of the University of Bonn, FR Germany.     

Duality in mathematics and linear and integer programming

Linear programming (LP) duality is examined in the context of other dualities in mathematics. The mathematical and economic properties of LP duality are discussed and its uses are considered. These mathematical and economic properties are then examined in relation to possible integer programming (IP) dualities. A number of possible IP duals are considered in this light and shown to capture some but not all desirable properties. It is shown that inherent in IP models are inequality and congruence constraints, both of which give on their own well-defined duals. However, taken together, no totally satisfactory dual emerges. The superadditive dual based on the Gomory and Chvátal functions is then described, and its properties are contrasted with LP duals and other IP duals. Finally, possible practical uses of IP duals are considered.The author is indebted to Professor H. B. Griffiths for many stimulating conversations on this topic.     

Ranking in integer linear fractional programming problems

An algorithm is developed which ranks the feasible solutions of an integer fractional programming problem in decreasing order of the objective function values.

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Zusammenfassung Es wird ein Algorithmus angegeben, der die zulässigen Lösungen eines ganzzahligen Quotientenprogrammes nach fallenden Zielfunktionswerten liefert.

     

Marginal values in mixed integer linear programming

For a given optimization problem, P, considered as a function of the data, its marginal values are defined as the directional partial derivatives of the value of P with respect to perturbations in that data. For linear programs, formulas for the marginal values were given by Mills, [10], and further developed by the current author [16]. In this paper, the marginal value formulas are extended to the case of mixed integer linear programming (MIP). As in ordinary linear programming, discontinuities in the value can occur, and the analysis here identifies them. This latter aspect extends previous work on continuity by the current author, [18], Geoffrion and Nauss, [5], Nauss, [11], and Radke, [12], and work on the value function of Blair and Jeroslow, [2]. Application is made to model formulation and to post-optimal analysis.Supported in part by the Air Force Office of Scientific Research, Grant # AFSOR-0271 to Rutgers University.     

Constrained integer linear fractional programming problem

An integer linear fractional programming problem, whose integer solution is required to satisfy any h out of given n sets of constraints has been discussed in this paper. Method for ranking and scanning all integer points has also been developed and a numerical illustration is included in support of theory.     

Complexity of linear relaxations in integer programming

Mathematical Programming - For a set X of integer points in a polyhedron, the smallest number of facets of any polyhedron whose set of integer points coincides with X is called the...     

A parallel integer linear programming algorithm

The solution of large scale integer linear programming models is generally dependent, in some way, upon the branch and bound technique, which can be quite time consuming. This paper describes a parallel branch and bound algorithm which achieves super linear efficiency in solving integer linear programming models on a multiprocessor computer. The algorithm is used to solve the Haldi and IBM test problems as well as a system design model.     

The use of linear programming duality in mixed integer programming

This paper is about the primal-dual relationship in a mixedinteger programming problem (MIP) in which integer variablesare binary. It shows how the primal-dual relationship of a linearprogramming problem (LP) can be used to advantage in MIPs. Thecentral idea is to look conceptually at the nature of all possibleLPs that arise from all possible settings for the discrete variablesin order to deduce general properties of the solution set. Afterdeveloping the relevant theory, we show the usefulness of thisaproach by applying it to three totally different problems.New results are derived for the method of least median of squaresin robust regression, the problem of rectilinear obnoxious-facilitylocation, and the problem of finding a fixed-size rectanglecontaining the minimum weight of points.     

On the greedy solution in integer linear programming

A version of the greedy method not using any knapsack relaxation of the integer programming problem is considered in this paper. It is based on a natural partial ordering of the vectors. Our aim is to determine a large class of problems where the greedy solution is always optimal. The results generalize some theorems of an early paper of Magazine, Nemhauser and Trotter and at the same time show a connection between two different notions of combinatorics: the greedy method and the Hilbert basis.

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Zusammenfassung In dieser Arbeit wird eine Version des Greedy-Algorithmus zur Lösung ganzzahliger linearer Optimierungsprobleme benutzt, die kein Rucksackproblem als Relaxation verwendet. Das Verfahren basiert auf der natürlichen partiellen Ordnung von Vektoren. Ziel der Arbeit ist es, eine möglichst große Problemklasse zu beschreiben, für die die Greedy-Lösung optimal ist. Die Ergebnisse verallgemeinern Sätze einer früheren Arbeit von Magazine, Nemhauser und Trotter und zeigen gleichzeitig einen Bezug zwischen zwei verschiedenen Gebieten der Kombinatorik auf: des Greedy-Verfahrens und von Hubert-Basen.

     

A linear integer programming bound for maximum-entropy sampling

 We introduce a new upper bound for the maximum-entropy sampling problem. Our bound is described as the solution of a linear integer program. The bound depends on a partition of the underlying set of random variables. For the case in which each block of the partition has bounded cardinality, we describe an efficient dynamic-programming algorithm to calculate the bound. For the very special case in which the blocks have no more than two elements, we describe an efficient algorithm for calculating the bound based on maximum-weight matching. This latter formulation has particular value for local-search procedures that seek to find a good partition. We relate our bound to recent bounds of Hoffman, Lee and Williams. Finally, we report on the results of some computational experiments. Received: September 27, 2000 / Accepted: July 26, 2001 Published online: September 5, 2002 Key words. experimental design – design of experiments – entropy – maximum-entropy sampling – matching – integer program – spectral bound – Fischer's inequality – branch-and-bound – dynamic programming Mathematics Subject Classification (2000): 52B12, 90C10 Send offprint requests to: Jon Lee Correspondence to: Jon Lee     

A linear algorithm for integer programming in the plane

We show that a 2-variable integer program, defined by m constraints involving coefficients with at most bits, can be solved with O(m+) arithmetic operations on rational numbers of size O().     

A two-stage approach for bi-objective integer linear programming

We present a new exact approach for solving bi-objective integer linear programs. The new approach employs two of the existing exact algorithms in the literature, including the balanced box and the $?$-constraint methods, in two stages. A computationally study shows that the new approach has three desirable characteristics. (1) It solves less single-objective integer linear programs. (2) Its solution time is significantly smaller. (3) It is competitive with the two-stage algorithm proposed by Leitner et al. (2016).     

An integer linear programming approach for a class of bilinear integer programs

We propose an Integer Linear Programming (ILP) approach for solving integer programs with bilinear objectives and linear constraints. Our approach is based on finding upper and lower bounds for the integer ensembles in the bilinear objective function, and using the bounds to obtain a tight ILP reformulation of the original problem, which can then be solved efficiently. Numerical experiments suggest that the proposed approach outperforms a latest iterative ILP approach, with notable reductions in the average solution time.     

An integer linear programming approach for approximate string comparison

We introduce a problem called maximum common characters in blocks (MCCB), which arises in applications of approximate string comparison, particularly in the unification of possibly erroneous textual data coming from different sources. We show that this problem is NP-complete, but can nevertheless be solved satisfactorily using integer linear programming for instances of practical interest. Two integer linear formulations are proposed and compared in terms of their linear relaxations. We also compare the results of the approximate matching with other known measures such as the Levenshtein (edit) distance.     

Exact augmented Lagrangian duality for mixed integer linear programming

We investigate the augmented Lagrangian dual (ALD) for mixed integer linear programming (MIP) problems. ALD modifies the classical Lagrangian dual by appending a nonlinear penalty function on the violation of the dualized constraints in order to reduce the duality gap. We first provide a primal characterization for ALD for MIPs and prove that ALD is able to asymptotically achieve zero duality gap when the weight on the penalty function is allowed to go to infinity. This provides an alternative characterization and proof of a recent result in Boland and Eberhard (Math Program 150(2):491–509, 2015, Proposition 3). We further show that, under some mild conditions, ALD using any norm as the augmenting function is able to close the duality gap of an MIP with a finite penalty coefficient. This generalizes the result in Boland and Eberhard (2015, Corollary 1) from pure integer programming problems with bounded feasible region to general MIPs. We also present an example where ALD with a quadratic augmenting function is not able to close the duality gap for any finite penalty coefficient.     

Lagrangean decomposition for integer nonlinear programming with linear constraints

We present a Lagrangean decomposition to study integer nonlinear programming problems. Solving the dual Lagrangean relaxation we have to obtain at each iteration the solution of a nonlinear programming with continuous variables and an integer linear programming. Decreasing iteratively the primal—dual gap we propose two algorithms to treat the integer nonlinear programming.This work was partially supported by CNPq and FINEP.