Projected Hessian Updating Algorithms for Nonlinearly Constrained Optimization

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...

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Bibliographic Details
Published in:SIAM journal on numerical analysis Vol. 22; no. 5; pp. 821 - 850
Main Authors: Nocedal, Jorge, Overton, Michael L.
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.10.1985
Subjects:
Algorithms
Approximation
Constrained optimization
Exact sciences and technology
Factorization
Hessian matrices
Jacobians
Lagrangian function
Least squares
Mathematics
Methods
Newtons method
Numerical analysis
Numerical analysis in abstract spaces
Numerical analysis. Scientific computation
Optimization
Sciences and techniques of general use
Optimization
Numerical analysis
ISSN:0036-1429, 1095-7170
Online Access:Get full text
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Summary: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 × (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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ISSN:0036-1429
1095-7170
DOI:10.1137/0722050