Duality and Sensitivity Analysis for Fractional Programs

Abstract

In this paper we consider algorithms, duality and sensitivity analysis for optimization problems, called fractional, whose objective function is the ratio of two real-valued functions. We discuss a procedure suggested by Dinkelbach for solving the problem, its relation to certain approaches via variable transformations, and a variant of the procedure that has convenient convergence properties. The duality correspondences that are developed do not require either differentiability or the existence of an optimal solution. The sensitivity analysis applies to linear fractional problems, even when they “solve” at an extreme ray, and includes a primal-dual algorithm for parametric right-hand-side analysis.

Sensitivity analysis in piecewise linear fractional programming problem with non-degenerate optimal solution

“…These results have been extended to the variations for both numerator and denominator of the objective function as well as with right-hand-side of the constraints. Then a primal-dual algorithm proposed to parametric right-hand-side analysis and this algorithm suggests a branch-bound method for integer linear programming [4]. An alternative procedure studied for multi-parametric sensitivity analysis in linear programming by the concept of a maximum volume in the tolerance region, which is bounded by a symmetrically rectangular parallelepiped and can be solved by a maximization problem [13].…”

Experiments with linear fractional problems

“…In this case x " = x 0 ptimai· K > s worth noting that Wagner and Yuan [10] related the two main approaches by showing that Martos's algorithm is equivalent to Charnes and Cooper's method in the sense that both algorithms lead to an identical sequence of pivoting operations. Bitran and Magnanti [1] have extended the connection between these approaches by relating them to generalized programming. No theoretical or empirical evidence has been given in the past indicating which of the several existing algorithms is preferred.…”

A New Approach for Solving Linear Fractional Programming Problems with Duality Concept

“…But to solve this sequence of problems, sometimes may need much iteration. Also some aspects concerning duality and sensitivity analysis in linear fractional program were discussed by Bitran and Magnant [3] and Singh [4], in his paper made a useful study about the optimality condition in fractional programming. Assuming the positivity of denominator of the objective function of LFP over the feasible region, Swarup [5] extended the wellknown simplex method to solve the LFP.…”