## Abstract

This paper is concerned with the problem min left brace f(x) vertical x an element of X right brace , where X is a convex subset of a linear space H, and f is a smooth real-valued function on H. The class of methods x//k// plus //1 equals P(x//k minus alpha //kg//k), is poposed, where P denotes projection on X with respect to a Hilbert space norm, g//k denotes the Frechet derivative of f at x//k with respect to another Hilbert norm, on H, and alpha //k is a positive scalar stepsize. It is then possible to match the first norm with the structure of X so that the projection operation is simplified while at the same time reserving the option to choose the second norm on the basis of approximations to the Hessian of f so as to attain a typically superlinear rate of convergence. The resulting methods are particularly attractive for large-scale problems with specially structured constraint sets such as optimal control and nonlinear multi-commodity network flow problems. The latter class of problems is discussed in some detail.

Original language | English (US) |
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Pages (from-to) | 936-964 |

Number of pages | 29 |

Journal | SIAM Journal on Control and Optimization |

Volume | 22 |

Issue number | 6 |

DOIs | |

State | Published - 1984 |

Externally published | Yes |

## ASJC Scopus subject areas

- Control and Optimization
- Applied Mathematics