### Abstract

We consider the problem of minimizing a smooth function of n variables subject to m smooth equality constraints. We begin by describing various approaches to Newton's method for this problem, with emphasis on the recent work of Goodman. This leads to the proposal of a Broyden-type method which updates an n multiplied by (n-m) matrix approximating a 'one-sided projected Hessian' of a Lagrangian function. This method is shown to converge Q-superlinearly. We also give a new short proof of the Boggs-Tolle-Wang necessary and sufficient condition for Q-superlinear convergence of a class of quasi-Newton methods for solving this problem. Finally, we describe an algorithm which updates an approximation to a 'two-sided projected Hessian', a symmetric matrix of order n-m which is generally positive definite near a solution. We present several new variants of this algorithm and show that under certain conditions they all have a local two-step Q-superlinear convergence property, even though only one set of gradients is evaluated per iteration. Numerical results are presented, indicating that the methods may be very useful in practice.

Original language | English (US) |
---|---|

Pages (from-to) | 821-850 |

Number of pages | 30 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 22 |

Issue number | 2 |

State | Published - Oct 1985 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics
- Computational Mathematics

### Cite this

*SIAM Journal on Numerical Analysis*,

*22*(2), 821-850.

**PROJECTED HESSIAN UPDATING ALGORITHMS FOR NONLINEARLY CONSTRAINED OPTIMIZATION.** / Nocedal, Jorge; Overton, Michael L.

Research output: Contribution to journal › Article

*SIAM Journal on Numerical Analysis*, vol. 22, no. 2, pp. 821-850.

}

TY - JOUR

T1 - PROJECTED HESSIAN UPDATING ALGORITHMS FOR NONLINEARLY CONSTRAINED OPTIMIZATION.

AU - Nocedal, Jorge

AU - Overton, Michael L.

PY - 1985/10

Y1 - 1985/10

N2 - We consider the problem of minimizing a smooth function of n variables subject to m smooth equality constraints. We begin by describing various approaches to Newton's method for this problem, with emphasis on the recent work of Goodman. This leads to the proposal of a Broyden-type method which updates an n multiplied by (n-m) matrix approximating a 'one-sided projected Hessian' of a Lagrangian function. This method is shown to converge Q-superlinearly. We also give a new short proof of the Boggs-Tolle-Wang necessary and sufficient condition for Q-superlinear convergence of a class of quasi-Newton methods for solving this problem. Finally, we describe an algorithm which updates an approximation to a 'two-sided projected Hessian', a symmetric matrix of order n-m which is generally positive definite near a solution. We present several new variants of this algorithm and show that under certain conditions they all have a local two-step Q-superlinear convergence property, even though only one set of gradients is evaluated per iteration. Numerical results are presented, indicating that the methods may be very useful in practice.

AB - We consider the problem of minimizing a smooth function of n variables subject to m smooth equality constraints. We begin by describing various approaches to Newton's method for this problem, with emphasis on the recent work of Goodman. This leads to the proposal of a Broyden-type method which updates an n multiplied by (n-m) matrix approximating a 'one-sided projected Hessian' of a Lagrangian function. This method is shown to converge Q-superlinearly. We also give a new short proof of the Boggs-Tolle-Wang necessary and sufficient condition for Q-superlinear convergence of a class of quasi-Newton methods for solving this problem. Finally, we describe an algorithm which updates an approximation to a 'two-sided projected Hessian', a symmetric matrix of order n-m which is generally positive definite near a solution. We present several new variants of this algorithm and show that under certain conditions they all have a local two-step Q-superlinear convergence property, even though only one set of gradients is evaluated per iteration. Numerical results are presented, indicating that the methods may be very useful in practice.

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UR - http://www.scopus.com/inward/citedby.url?scp=0022145863&partnerID=8YFLogxK

M3 - Article

VL - 22

SP - 821

EP - 850

JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

SN - 0036-1429

IS - 2

ER -