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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Veröffentlicht in:	SIAM journal on numerical analysis Jg. 22; H. 5; S. 821 - 850
Hauptverfasser:	Nocedal, Jorge, Overton, Michael L.
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Philadelphia, PA Society for Industrial and Applied Mathematics 01.10.1985
Schlagworte:	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
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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 × (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.
AbstractList	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. 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 $n \times (n - m)$ 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.
Author	Nocedal, Jorge Overton, Michael L.
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Cites_doi	10.1093/imamat/12.3.223 10.1007/BF01580366 10.1007/BF01588967 10.1137/1019005 10.1007/978-3-642-48320-2 10.1137/0721051 10.1007/BF01580395 10.1007/BFb0120946 10.1007/BF00928472 10.1007/BF00932842 10.1007/BF01585529 10.1007/BF01588224 10.1007/BF02592223 10.1016/B978-0-12-468660-1.50007-4 10.1007/BF01580651 10.1007/978-3-642-68874-4_12 10.1137/0320014 10.1007/BF00933161
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References	Tanabe Kunio (R35) 1981; 3 Dennis, Jr. John E. (R9) 1983 R21 Murray W. (R27) 1969 R23 R25 Bertsekas Dimitri P. (R1) 1982 R26 Bartholomew-Biggs M. C. (R2) 1982 Biggs M. C. (R3) 1972 R4 Himmelblau D. (R24) 1972 R5 R6 Fletcher R. (R11) 1981 R7 R8 Gill P. E. (R18) 1974 R32 R31 Fletcher R. (R10) 1974 R33 R14 R36 R16 R15 R37 Ortega J. M. (R30) 1970 R17 R19
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Snippet	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... We consider the problem of minimizing a smooth function of $n$ variables subject to $m$ smooth equality constraints. We begin by describing various approaches...
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SubjectTerms	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
Title	Projected Hessian Updating Algorithms for Nonlinearly Constrained Optimization
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