A new descent algorithm with curve search rule for unconstrained minimization

In the paper we present a new descent algorithm with curve search rule for unconstrained minimization problems. At each iteration, the next iterative point is determined by means of a curve search rule. It is particular that the search direction and the step size is determined simultaneously at each iteration of the new algorithm. Similarly to conjugate gradient methods, the algorithm avoids the computation and storage of some matrices associated with the Hessian of objective functions. It is suitable to solve large scale minimization problems. Numerical experiments show that our algorithm is effective in practical computation.

I. Introduction

Let be an n-dimensional Euclidean space and is a continuously differentiable function. Unconstrained minimization problem (UP) is to find the minimal point of over , denoted by

$$\min f(x),\quad x\in R^{n}.$$ Most of the well-known methods for solving (UP) take the form $$x_{k+1}=x_{k}+\alpha_{k}d_{k}, k=1,2, \ldots, \eqno\hbox{(1)}$$ where is a search direction of at and is a positive step-size. If is the current point, we denote by by and by , respectively.