On the uniform nonsingularity of matrices of search directions and the rate of convergence in minimization algorithms

A convergent minimization algorithm made up of repetitive line searches is considered in ⁿ. It is shown that the uniform nonsingularity of the matrices consisting ofn successive normalized search directions guarantees a speed of convergence which is at leastn-step Q-linear. Consequences are given for multistep methods, including Powell's 1964 procedure for function minimization without calculating derivatives as well as Zangwill's modifications of this procedure.The authors wish to thank the Namur Department of Mathematics, especially its optimization group, for many discussions and encouragement. They also thank the reviewers for many helpful suggestions.