A Method of Conjugate Directions for Linearly Constrained Nonlinear Programming Problems

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 $\{ {x_j } \}$ generated by this method is a stationary point. If $\{ {x_j } \}$ 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 $\{ {x_j } \}$ 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 $\{ {y_j } \}$ from the sequence $\{ {x_j } \}$ which converges faster to z in the sense that $\| {y_j - z} \|\| {x_j - z} \|^{ - 1} \to 0$ as $j \to \infty $.