A quadratically convergent method for minimizing a sum of euclidean norms

We consider the problem of minimizing a sum of Euclidean norms. {Mathematical expression} here the residuals {r_i(x)} are affine functions from Rⁿ to R¹ (n≥1≥2, m>-2). This arises in a number of applications, including single-and multi-facility location problems. The function F is, in general, not differentiable at x if at least one r_i(x) is zero. Computational methods described in the literature converge quite slowly if the solution is at such a point. We present a new method which, at each iteration, computes a direction of search by solving the Newton system of equations, projected, if necessary, into a linear manifold along which F is locally differentiable. A special line search is used to obtain the next iterate. The algorithm is closely related to a method recently described by Calamai and Conn. The new method has quadratic convergence to a solution x under given conditions. The reason for this property depends on the nature of the solution. If none of the residuals is zero at^*x, then F is differentiable at^*x and the quadratic convergence follows from standard properties of Newton's method. If one of the residuals, say r_i^*x), is zero, then, as the iteration proceeds, the Hessian of F becomes extremely ill-conditioned. It is proved that this illconditioning, instead of creating difficulties, actually causes quadratic convergence to the manifold (xℛr_i(x)=0}. If this is a single point, the solution is thus identified. Otherwise it is necessary to continue the iteration restricted to this manifold, where the usual quadratic convergence for Newton's method applies. If several residuals are zero at^*x, several stages of quadratic convergence take place as the correct index set is constructed. Thus the ill-conditioning property accelerates the identification of the residuals which are zero at the solution. Numerical experiments are presented, illustrating these results.