Siegfried Schaible

An Algorithm for Generalized Fractional Programs

An algorithm is suggested that finds the constrained minimum of the maximum of finitely many ratios. The method involves a sequence of linear (convex) subproblems if the ratios are linear (convex-concave). Convergence results as well as rate of convergence results are derived. Special consideration is given to the case of (a) compact feasible regions and (b) linear ratios.

Duality in generalized fractional programming via Farkas' lemma

For fractional programs involving several ratios in the objective function, a dual is introduced with the help of Farkas lemma. Often the dual is again a generalized fractional program. Duality relations are established under weak assumptions. This is done in both the linear case and the nonlinear case. We show that duality can be obtained for these nonconvex programs using only a basic result on linear (convex) inequalities.

Duality in Fractional Programming: A Unified Approach

This paper presents a unified method for obtaining duality results for concave-convex fractional programs. We obtain these results by transforming the original nonconvex programming problem into an equivalent convex program. Known results by several authors are related to each other. Moreover, we prove additional duality theorems, in particular, converse duality theorems for nondifferentiable as well as quadratic fractional programs.

Duality in generalized linear fractional programming

We consider a generalization of a linear fractional program where the maximum of finitely many linear ratios is to be minimized subject to linear constraints. For this Min-Max problem, a dual in the form of a Max-Min problem is introduced and duality relations are established.

Bibliography in fractional programming

ZusammenfassungEs wird eine Bibliographie zur Quotientenprogrammierung veröffentlicht, die 551 Titel enthält. Es wurde versucht, alle Beiträge zu diesem Gebiet der nichtlinearen Programmierung zu berücksichtigen, das nun seit mehr als 45 Jahren erforscht wird.AbstractA 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.

A note on an algorithm for generalized fractional programs

We present a modification of an algorithm recently suggested by the same authors in this journal (Ref. 1). The speed of convergence is improved for the same complexity of computation.

Quasiconvex, pseudoconvex, and strictly pseudoconvex quadratic functions

The purpose of this paper is twofold. Firstly, criteria for quasiconvex and pseudoconvex quadratic functions in nonnegative variables of Cottle, Ferland, and Martos are derived by specializing criteria proved by the author. We do not make use of the concept of positive subdefinite matrices. Instead, we are specializing criteria that were derived for quadratic functions on arbitrary convex sets to the special case of quadratic functions in nonnegative variables. The second purpose of this paper is to present several new criteria involving also strictly pseudoconvex quadratic functions.

Quasi-concavity and pseudo-concavity of cubic functions

Generally, hard work must be done to prove quasi(pseudo-)concavity using the definitions [ 1 , 6 ] . However, criteria for various classes of functions can easily be derived if functions are considered as composed functions (see [7, 10, 11, 12] ). In this way also cubic functions can be investigated [10] . It suffices to observe quasiand pseudo-concave functions since a cubic function is strictly quasi-concave if and only if it is quasi-concave [ 8 ]. Consider K(x) = Q(x). l (x) , where Q(x) = xT A x + bTx + ~/ (where A is a real symmetric n Xn matrix, b ~ R n, / real) and l(x) = cTx + (c ~ R n , a real). Using an appropriate affine transformation x = Py + v, Q(x) reduces either to

On the convexifiability of pseudoconvex C2-functions

We present new criteria that characterize functions which are convex transformable by a suitable strictly increasing function. We concentrate on twice continuously differentiable pseudoconvex and strictly pseudoconvex functions, and derive conditions which are both necessary and sufficient for these functions to be convex transformable.

Multi-Ratio Fractional Programing — A Survey

Nonlinear programming problems are considered where the objective function involves several ratios. We review recent results concerning three classes of multi-ratio fractional programs: 1) maximization of a sum of ratios, 2) maximization of the smallest of several ratios, and 3) multiobjective fractional programs. In addition to these results open problems are addressed as well.