The optimization problem is to minimize $f(\mathbf {x} )$ where $\mathbf {x}$ is a vector in $R^{n}$, and $f$ is a differentiable scalar function. There are no constraints on the values that $\mathbf {x}$ can take.

The algorithm begins at an initial estimate for the optimal value $\mathbf {x}_{0}$ and proceeds iteratively to get a better estimate at each stage.

The search direction $p_k$ at stage $k$ is given by the solution of the analogue of the Newton equation

$$B_{k}\mathbf {p} _{k}=-\nabla f(\mathbf {x}_{k}),$$

where $B_{k}$ is an approximation to the Hessian matrix, which is updated iteratively at each stage, and $\nabla f(\mathbf {x} _{k})$ is the gradient of the function evaluated at $x_k$. A line search in the direction $p_k$ is then used to find the next point $x_{k+1}$ by minimising

$$f(\mathbf {x}_{k}+\alpha \mathbf {p}_{k}),$$

over the scalar $\alpha > 0$.