A bibliography in fractional programming is provided which contains 551 references. It was attempted to include all publications in this area of nonlinear programming as they have appeared in more than 45 years now.

Bibliography in fractional programming

A nonlinear fractional programming approach for environmental–economic power dispatch

Light from an incoherently radiating object and background light are focused onto a photosensitive mosaic, the currents from whose elements constitute the data on which is based a least-squares linear estimate of the radiance at points in the object. By comparison of the mean-square error with that given by Wiener filtering theory, the equivalent noise spectral density for use in the latter is shown to consist of a shot-noise term and a term due to the random fluctuations of the incoherent light; the former predominates under most circumstances. Turbulent distortion of the image after passage of the rays through a random-phase screen is also treated from this standpoint.

Linear Restoration of Incoherently Radiating Objects

A generalized fractional programming problem is defined as the problem of minimizing a nonlinear function, defined as the maximum of several ratios of functions on a feasible domain. In this paper, we propose new methods based on the method of centers, on the proximal point algorithm and on the idea of bundle methods, for solving such problems. First, we introduce proximal point algorithms, in which, at each iteration, an approximate prox-regularized parametric subproblem is solved inexactly to obtain an approximate solution to the original problem. Based on this approach and on the idea of bundle methods, we propose implementable proximal bundle algorithms, in which the objective function of the last mentioned prox-regularized parametric subproblem is replaced by an easier one, typically a piecewise linear function. The methods deal with nondifferentiable nonlinearly constrained convex minimax fractional problems. We prove the convergence, give the rate of convergence of the proposed procedures and present numerical tests to illustrate their behavior.

Proximal Bundle Algorithms for Nonlinearly Constrained Convex Minimax Fractional Programs

We propose new duality results for generalized fractional programs (GFP) for a wide class of problems, not limited only to the convex case. Our approach does not use Lagrangian duality, but only an...

Duality Results and Dual Bundle Methods Based on the Dual Method of Centers for Minimax Fractional Programs

where xeE , S = {xeE :0 0, (ii) B is symmetric and positive definite, (iii) # > 0. This problem arises in the maximization of the signal-to-noise ratio in a spectral filter for infrared detectors (see [7]). The term 2 represents the incoming signal and + ,B is the variance of the background signal. The constraint S reflects the fact that the filter can transmit no more than 100 % and not less than 0 % of the total energy. A paper detailing the physics of the problem is in preparation. Although various general-purpose algorithms have been developed which may be applied to obtain solutions to this problem, the special nature of the objective function and the constraints can be exploited to obtain more efficient solution techniques. Two related algorithms are presented here and are both based on a parametric analysis of the problem. The second method is based on the first and offers some computational advantages that the first does not possess. The derivation of the second method involves the notion of the "conjugate function" [6] in a unique manner and should be of interest in itself since the practical applications of this notion have heretofore been extremely limited.

James E. Falk

Papers

Maximization of Signal-to-Noise Ratio in an Optical Filter