Fractional Programming with Homogeneous Functions

Abstract

This paper extends the well known results for linear fractional programming to the class of programming problems involving the ratio of nonlinear functionals subject to nonlinear constraints, where the constraints are homogeneous of degree one and the functionals are homogeneous of degree one to within a constant. Two rather general auxiliary problems are developed, and the relations between the solutions of the auxiliary problems and the solutions of the original problem are codified. Applications of the results for specific problems are also presented.

Fixed point quasiconvex subgradient method

“…The Cobb-Douglas production efficiency problem was introduced by Bradley and Frey [3]. Hu, Yang, and Sim proposed an algorithm for solving this problem by regarding it as a constrained quasiconvex optimization problem [11].…”

“…Therefore, these problems can be dealt with as constrained quasiconvex optimizations. Here, we will examine the numerical behaviors of the existing and proposed algorithms when they are applied to the Cobb-Douglas production efficiency problem [3,Problem (3.13)], [11,Problem (6.1)], [35, Section 1.7], which is an instance of a fractional programming and constrained quasiconvex optimization problem. Furthermore, the demand for techniques to solve optimization problems is nowadays not only limited to convex objectives.…”

Experiments with linear fractional problems

“…This approach has been extended to the nonlinear versions of (P) by Bradley and Frey [3] and Schaible [8]. The second approach solves a sequence of linear problems or at least one pivot step of each linear program over the original feasible set by updating the objective function.…”

Duality and Sensitivity Analysis for Fractional Programs

“…For an approach to the fractional problem that generalizes the Charnes and Cooper algorithm and applies when n(x) and d(x) are homogeneous of degree one to within a constant see Bradley and Frey [6].…”

“…The new value for {N is computed by (6) and then a usual dual simplex pivot is made in row i (if every constraint coefficient in row i is nonnegative, linear programming duality theory [10] shows that the problem is infeasible for > 00 and the procedure terminates).…”