A robust algorithm that minimizes L-function in a finite number of steps and rapidly minimizes general functions

A new conjugate-gradient algorithm which minimizes a function of n variables is given. The algorithm performs n orthogonal searches in each stage and hence has the property that it is robust, i.e. it will not easily fail on functions which have a large number of variables (n>10) nor on functions which have "ridge like" properties. A general class of functions called L-functions, which includes the class of quadratic functions as a special case, is defined, and it is shown that the algorithm has the property that it will converge to the minimum of a L-function in n (or less) 1-dimensional, minimizations and (n-1) (or less) 1-dimensional pseudo-minimizations. Numerical experiments are included for systems of 2nd to 40th order, and based on these experiments (assuming the gradients are calculated numerically) the new algorithm appears to be more robust than Powell's [10], Fletcher-Powell's [11], and Jacobson-Oksman's [14] methods, faster than Rosenbrock's [9] method, and especially effective on high dimensional problems.