A generalized fractional programming problem is specified as a nonlinear program where a nonlinear function defined as the maximum over several ratios of functions is to be minimized on a feasible domain of źn. The purpose of this paper is to outline basic approaches and basic types of algorithms available to deal with this problem and to review their convergence analysis. The conclusion includes results and comments on the numerical efficiency of these algorithms.

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Algorithms for generalized fractional programming

The problems of designing and implementing LSI systems for the processing of 2-D digital data, such as images or geophone arrays, are reviewed and discussed. This discussion encompasses both FIR and IIR digital filters and with respect to the latter, the issues of stability testing and filter stabilization are also considered. Techniques are also presented whereby such filtering can be accomplished using either 1 or 2-D LSI systems.

Two-dimensional digital filtering

This bibliography of fractional programming is a continuation of five previous bibliographies by the author (Pure Appl. Math. Sci. (India), Vol. XIII, No. 1–2, 35–69, March (1981); ibid. Vol. XVII, No. 1–2, 87–102, March (1983); ibid. XXII, No. 1–2, 109–122, September (1985); Optimization 23(1992)1, 53–71; ibid. 45(1999) 1–4, 343–367). This compilation lists, in alphabetical order by the name of the first author, 491 papers dealing with fractional programming and its applications. This covers mainly the period 1997–2005 but it also includes some references published up to 1997 which were not included in the previous bibliographies or which were mentioned as ‘to appear’ in the five bibliography. In compiling this list we used Mathematical Reviews, Zentralblatt fur Mathematik, Referativnyi Zhurnal (Matematika) and Current Papers on Computers & Control. The papers are either published in some form (in technical journals or as internal reports) or are available only as typewritten manuscripts (for example, as...

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A sixth bibliography of fractional programming

Nonsmooth optimization problems are divided into two categories. The first is composite nonsmooth problems where the generalized gradient can be approximated by information available at the current point. The second is basic nonsmooth problems where the generalized gradient must be approximated using information calculated at previous iterates.

An algorithm for composite nonsmooth optimization problems

Many problems in multi-view geometry, when posed as minimization of the maximum reprojection error across observations, can be solved optimally in polynomial time. We show that these problems are instances of a convex-concave generalized fractional program. We survey the major solution methods for solving problems of this form and present them in a unified framework centered around a single parametric optimization problem. We propose two new algorithms and show that the algorithm proposed by Olsson et al. [21] is a special case of a classical algorithm for generalized fractional programming. The performance of all the algorithms is compared on a variety of datasets, and the algorithm proposed by Gugat [12] stands out as a clear winner. An open source MATLAB toolbox that implements all the algorithms presented here is made available.

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Fast algorithms for L ∞ problems in multiview geometry

The version of the “differential correction algorithm” that is most used at the present time is a modification of the original version, perhaps because it has been proved that the modified version ...

The Differential Correction Algorithm for Rational $\ell _\infty $-Approximation

Given a set of function values f(xt) (t=1,2,...,N), we consider the problem of calculating the rational function R(x)=P(x)/Q(x) that minimizes the quantity 

$$\begin{gathered}\max |f(xt) - R(xt)|, \hfill \\t \hfill \\\end{gathered}$$

where P(x) and Q(x) are polynomials of prescribed degrees. To solve this problem Cheney and Loeb [2] proposed a "differential correction algorithm", ODC say, but in a subsequent paper [3] they modified their algorithm, and now the modified algorithm, DC say, is nearly always used in place of ODC. The purpose of this paper is to direct attention back to the original algorithm, because in practice ODC seems to be much better.

M. J. D. Powell

Papers

The Differential Correction Algorithm for Rational $\ell _\infty $-Approximation

The differential correction algorithm for rational L∞ approximation