On duality in linear fractional programming

In this paper, a dual of a given linear fractional program is defined and the weak, direct and converse duality theorems are proved. Both the primal and the dual are linear fractional programs. This duality theory leads to necessary and sufficient conditions for the optimality of a given feasible solution. A unmerical example is presented to illustrate the theory in this connection. The equivalence of Charnes and Cooper dual and Dinkelbach’s parametric dual of a linear fractional program is also established.     

Comparison of duality models in fractional linear programming

Summary This paper deals with the duality models in fractional linear programming presented in the last years bySwarup, Kaka, Sharma andSwarup and other authors.

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Zusammenfassung Der Aufsatz befaßt sich mit Dualitätsmodellen für Linear Fractional Programming, die in den letzten Jahren vonSwarup, Kaka, Sharma undSwarup sowie von anderen Autoren angegeben wurden.

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This work was sponsored by the Grant No. A 7329 from the National Research Council of Canada.     

On duality concepts in fractional programming

The paper (Part I) describes an approach to duality in fractional programming on the basis of another kind of conjugate functions. The connections to some duality concepts (the Lagrange-duality and duality concepts of Craven and Schaible) are investigated and some new proofs of strong duality theorems are given.     

On duality in multiple objective linear programming

In this paper we present two approaches to duality in multiple objective linear programming. The first approach is based on a duality relation between maximal elements of a set and minimal elements of its complement. It offers a general duality scheme which unifies a number of known dual constructions and improves several existing duality relations. The second approach utilizes polarity between a convex polyhedral set and the epigraph of its support function. It leads to a parametric dual problem and yields strong duality relations, including those of geometric duality.     

On strong duality in linear copositive programming

Journal of Global Optimization - The paper is dedicated to the study of strong duality for a problem of linear copositive programming. Based on the recently introduced concept of the set of...     

Linear fractional programming and duality

This paper presents a dual of a general linear fractional functionals programming problem. Dual is shown to be a linear programming problem. Along with other duality theorems, complementary slackness theorem is also proved. A simple numerical example illustrates the result.     

Conjugate duality in generalized fractional programming

The concepts of conjugate duality are used to establish dual programs for a class of generalized nonlinear fractional programs. It is now known that, under certain restrictions, a symmetric duality exists for generalized linear fractional programs. In this paper, we establish this symmetric duality for the nonlinear case.     

On robust duality for fractional programming with uncertainty data

In this paper, we present a duality theory for fractional programming problems in the face of data uncertainty via robust optimization. By employing conjugate analysis, we establish robust strong duality for an uncertain fractional programming problem and its uncertain Wolfe dual programming problem by showing strong duality between the deterministic counterparts: robust counterpart of the primal model and the optimistic counterpart of its dual problem. We show that our results encompass as special cases some programming problems considered in the recent literature. Moreover, we also show that robust strong duality always holds for linear fractional programming problems under scenario data uncertainty or constraint-wise interval uncertainty, and that the optimistic counterpart of the dual is tractable computationally.     

On duality theorems for a nondifferentiable minimax fractional programming

The optimality conditions of [Lai et al. (J. Math. Anal. Appl. 230 (1999) 311)] can be used to construct two kinds of parameter-free dual models of nondifferentiable minimax fractional programming problems which involve pseudo-/quasi-convex functions. In this paper, the weak duality, strong duality, and strict converse duality theorems are established for the two dual models.     

Robustness and duality in linear programming

In this paper, we consider a linear program in which the right hand sides of theconstraints are uncertain and inaccurate. This uncertainty is represented byintervals, that is to say that each right hand side can take any value in itsinterval regardless of other constraints. The problem is then to determine arobust solution, which is satisfactory for all possible coefficient values.Classical criteria, such as the worst case and the maximum regret, are appliedto define different robust versions of the initial linear program. Morerecently, Bertsimas and Sim have proposed a new model that generalizes the worstcase criterion. The subject of this paper is to establish the relationshipsbetween linear programs with uncertain right hand sides and linear programs withuncertain objective function coefficients using the classical duality theory. Weshow that the transfer of the uncertainty from the right hand sides to theobjective function coefficients is possible by establishing new dualityrelations. When the right hand sides are approximated by intervals, we alsopropose an extension of the Bertsimas and Sim's model and we show that themaximum regret criterion is equivalent to the worst case criterion.     

Vector-valued lagrangian and multiobjective fractional programming duality

A class of multiobjective fractional programming problems is considered and duality results are established in terms of properly efficient solutions of the primal and dual programs. Further a vector-valued ratio type Lagrangian is introduced and certain vector saddlepoint results are presented.     

Second order duality for minmax fractional programming

In the present paper, two types of second order dual models are formulated for a minmax fractional programming problem. The concept of η-bonvexity/generalized η-bonvexity is adopted in order to discuss weak, strong and strict converse duality theorems. The research of Z. Husain is supported by the Department of Atomic Energy, Government of India, under the NBHM Post-Doctoral Fellowship Program No. 40/9/2005-R&D II/1739.     

Symmetric duality for disjunctive programming with absolute value functionals

In this paper we develop a complete duality theory for a couple of disjunctive linear programming problems with absolute value functionals. The pair of dual problems constructed has no duality gap, and may be considered as a generalization of the duality theory for convex programming.     

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.     

Strong duality for inexact linear programming

In this paper, we apply the Dubovitskii-Milyutin approach to derive strong duality theorems for inexact linear programming problems. Inexact linear programming deals with the standard linear problem in which the data is not well known and it is supposed to lie in certain given convex sets. The case of parametric dependence of the data is particularly analyzed and relations with semi-infinite and

semi-definite programming are also commented.     

Generalized fractional programming: Optimality and duality theory

The generalized fractional programming problem with a finite number of ratios in the objective is studied. Optimality and duality results are established, some with the help of an auxiliary problem and some directly. Convexity and stability of the auxiliary problem play a key role in the latter part of the paper.The authors are grateful to an unknown referee for suggesting the statement of Theorem 3.3.