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.     

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.     

An integral transformation for integer programming problems

It is shown that every integer programming problem can be transformed into an equivalent integer program with free variables in polynomial time. The transformation is advantageous because the equivalent problem it generates can be solved very easily in some restricted cases.     

A linear programming approach to test efficiency in multi-objective linear fractional programming problems

In a multi-objective linear fractional programming problem (MOLFPP), it is often useful to check the efficiency of a given feasible solution, and if the solution is efficient, it is useful to check strong or weak efficiency. In this paper, by applying a geometrical interpretation, a linear programming approach is achieved to test weak efficiency. Also, in order to test strong efficiency for a given weakly efficient point, a linear programming approach is constructed.     

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.     

Bicriteria linear fractional programming

As a step toward the investigation of the multicriteria linear fractional program, this paper provides a thorough analysis of the bicriteria case. It is shown that the set of efficient points is a finite union of linearly constrained sets and the efficient frontier is the image of a finite number of connected line segments of efficient points. A simple algorithm using only one-dimensional parametric linear programming techniques is developed to evaluate the efficient frontier.This research was partially supported by NRC Research Grant No. A4743. The authors wish to thank two anonymous referees for their helpful comments on an earlier draft of this paper.     

On a pair of nonlinear mixed integer programming problems

We consider maximin and minimax nonlinear mixed integer programming problems which are nonsymmetric in duality sense. Under weaker (pseudo-convex/pseudo-concave) assumptions, we show that the supremum infimum of the maximin problem is greater than or equal to the infimum supremum of the minimax problem. As a particular case, this result reduces to the weak duality theorem for minimax and symmetric dual nonlinear mixed integer programming problems. Further, this is used to generalize available results on minimax and symmetric duality in nonlinear mixed integer programming.     

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.

     

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.     

Weighting method for bi-level linear fractional programming problems

In this paper we consider the solution of a bi-level linear fractional programming problem (BLLFPP) by weighting method. A non-dominated solution set is obtained by this method. In this article decision makers (DMs) provide their preference bounds to the decision variables that is the upper and lower bounds to the decision variables they control. We convert the hierarchical system into scalar optimization problem (SOP) by finding proper weights using the analytic hierarchy process (AHP) so that objective functions of both levels can be combined into one objective function. Here the relative weights represent the relative importance of the objective functions.     

Minimizing the sum of a linear and a linear fractional function applying conic quadratic representation: continuous and discrete problems

This paper tries to minimize the sum of a linear and a linear fractional function over a closed convex set defined by some linear and conic quadratic constraints. At first, we represent some necessary and sufficient conditions for the pseudoconvexity of the problem. For each of the conditions, under some reasonable assumptions, an appropriate second-order cone programming (SOCP) reformulation of the problem is stated and a new applicable solution procedure is proposed. Efficiency of the proposed reformulations is demonstrated by numerical experiments. Secondly, we limit our attention to binary variables and derive a sufficient condition for SOCP representability. Using the experimental results on random instances, we show that the proposed conic reformulation is more efficient in comparison with the well-known linearization technique and it produces more eligible cuts for the branch and bound algorithm.     

A redundancy detection algorithm for fuzzy stochastic multi-objective linear fractional programming problems

The computational complexity of linear and nonlinear programming problems depends on the number of objective functions and constraints involved and solving a large problem often becomes a difficult task. Redundancy detection and elimination provides a suitable tool for reducing this complexity and simplifying a linear or nonlinear programming problem while maintaining the essential properties of the original system. Although a large number of redundancy detection methods have been proposed to simplify linear and nonlinear stochastic programming problems, very little research has been developed for fuzzy stochastic (FS) fractional programming problems. We propose an algorithm that allows to simultaneously detect both redundant objective function(s) and redundant constraint(s) in FS multi-objective linear fractional programming problems. More precisely, our algorithm reduces the number of linear fuzzy fractional objective functions by transforming them in probabilistic–possibilistic constraints characterized by predetermined confidence levels. We present two numerical examples to demonstrate the applicability of the proposed algorithm and exhibit its efficacy.     

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).     

Cutting plane algorithms for the inverse mixed integer linear programming problem

We present cutting plane algorithms for the inverse mixed integer linear programming problem (InvMILP), which is to minimally perturb the objective function of a mixed integer linear program in order to make a given feasible solution optimal.     

Ranking in quadratic integer programming problems

The present paper develops an algorithm for ranking the integer feasible solutions of a quadratic integer programming (QIP) problem. A linear integer programming (LIP) problem is constructed which provides bounds on the values of the objective function of the quadratic problem. The integer feasible solutions of this related integer linear programming problem are systematically scanned to rank the integer feasible solutions of the quadratic problem in non-decreasing order of the objective function values. The ranking in the QIP problem is useful in solving a nonlinear integer programming problem in which some other complicated nonlinear restrictions are imposed which cannot be included in the simple linear constraints of QIP, the objective function being still quadratic.     

The tolerance approach in multiobjective linear fractional programming

When solving a multiobjective programming problem by the weighted sum approach, weights represent the relative importance associated to the objectives. As these values are usually imprecise, it is important to analyze the sensitivity of the solution under possible deviations on the estimated values. In this sense, the tolerance approach provides a direct measure of how weights may vary simultaneously and independently from their estimated values while still retaining the same efficient solution. This paper provides an explicit expression to the maximum tolerance on weights in a multiobjective linear fractional programming problem when all the denominators are equal. An application is also presented to illustrate how the results may help the decision maker to choose a most satisfactory solution in a production problem.     

Strong duality for robust minimax fractional programming problems

We develop a duality theory for minimax fractional programming problems in the face of data uncertainty both in the objective and constraints. Following the framework of robust optimization, we establish strong duality between the robust counterpart of an uncertain minimax convex–concave fractional program, termed as robust minimax fractional program, and the optimistic counterpart of its uncertain conventional dual program, called optimistic dual. In the case of a robust minimax linear fractional program with scenario uncertainty in the numerator of the objective function, we show that the optimistic dual is a simple linear program when the constraint uncertainty is expressed as bounded intervals. We also show that the dual can be reformulated as a second-order cone programming problem when the constraint uncertainty is given by ellipsoids. In these cases, the optimistic dual problems are computationally tractable and their solutions can be validated in polynomial time. We further show that, for robust minimax linear fractional programs with interval uncertainty, the conventional dual of its robust counterpart and the optimistic dual are equivalent.     

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.     

Approximation algorithms for combinatorial fractional programming problems

We are concerned with a combinatorial optimization problem which has the ratio of two linear functions as the objective function. This type of problems can be solved by an algorithm that uses an auxiliary problem with a parametrized linear objective function. Because of its combinatorial nature, however, it is often difficult to solve the auxiliary problem exactly. In this paper, we propose an algorithm which assumes that the auxiliary problems are solved only approximately, and prove that it gives an approximate solution to the original problem, of which the accuracy is at least as good as that of approximate solutions to the auxiliary problems. It is also shown that the time complexity is bounded by the square of the computation time of the approximate algorithm for the auxiliary problem. As an example of the proposed algorithm, we present a fully polynomial time approximation scheme for the fractional 0–1 knapsack problem.