A subspace limited memory quasi-Newton algorithm for large-scale nonlinear bound constrained optimization.

*(English)*Zbl 0886.65065The following nonlinear programming problem with simple bounds on variables

$$\text{minimize}\phantom{\rule{4.pt}{0ex}}f\left(x\right)\phantom{\rule{1.em}{0ex}}\text{subject}\phantom{\rule{4.pt}{0ex}}\text{to}\phantom{\rule{4.pt}{0ex}}\ell \le x\le u$$

is considered. The objective function $f\left(x\right)$ is assumed to be twice continuously differentiable, $\ell $ and $u$ are given bound vectors in ${\mathbb{R}}^{n}$, and $n$ is the number of variables, which is assumed to be large.

The given subspace limited memory quasi-Newton algorithm does not need to solve any subproblems. The search direction of the algorithm consists of three parts: a subspace quasi-Newton direction, and two subspace gradient and modified gradient directions. The global convergence of the method is proved and some numerical results are given.

Reviewer: H.Benker (Merseburg)

##### MSC:

65K05 | Mathematical programming (numerical methods) |

90C06 | Large-scale problems (mathematical programming) |

90C30 | Nonlinear programming |