A Method of Conjugate Directions for Linearly Constrained Nonlinear Programming Problems

• Published in: SIAM journal on numerical analysis Vol. 12; no. 3; pp. 273 - 303
• Main Author: Ritter, Klaus
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.06.1975
• Subjects: Algorithms; Hessian matrices; Lipschitz condition; Mathematical functions; Mathematical vectors; Matrices; Nonlinear programming; Optimal solutions; Real numbers; Steepest descent method; Taylors theorem
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/0712024

An iterative method is described for the minimization of a continuously differentiable function F(x) of n variables subject to linear inequality constraints. Without any assumptions on second order derivatives it is shown that every cluster point of the sequence {xj} generated by this method is a stationary point. If {xj} has a cluster point z such that F(x) is twice continuously differentiable in some neighborhood of z and the Hessian matrix of F(x) has certain properties, then {xj} converges to z and the convergence is (n - p)-step superlinear, where p is the number of constraints which are active for z. Furthermore, a simple procedure is given for deriving a new sequence {yj} from the sequence {xj} which converges faster to z in the sense that |yj- z| |xj- z|-1→ 0 as j → ∞.