Optimization of a linear fractional function on a hypersphere of an affine space

Abstract

In this paper, we consider the following nonlinear programming problem:

$$max[min]\{ f(x) = (c^T \cdot x + c_0 )/(d^T \cdot x + d_0 )|x \in L\} ,$$

in the feasible region

$$L = \{ x \in R^n |A \cdot x = b,l(x) \leqslant 0\} ;$$

here,c andd are vectors ofR ⁿ;c ₀,d ₀ are real constants;A is anm×n matrix of rankm; b is a vector ofR ^m;l(x)=0 is the equation of a bounded hypersurface inR ⁿ. We assume thatd ^T·x+d ₀≠0 in L. We study the case where

$$l(x) = |x - e|^2 - r^2 ,$$

wherer≠0 ande is the vector with all the components equal to 1. We obtain a simple explicit solution, and we illustrate the resulting algorithm.