# The convergence of variable metric matrices in unconstrained optimization

@article{Ge1983TheCO, title={The convergence of variable metric matrices in unconstrained optimization}, author={Renpu Ge and M. J. D. Powell}, journal={Mathematical Programming}, year={1983}, volume={27}, pages={123-143} }

It is proved that, if the DFP or BFGS algorithm with step-lengths of one is applied to a functionF(x) that has a Lipschitz continuous second derivative, and if the calculated vectors of variables converge to a point at which ∇F is zero and ∇2F is positive definite, then the sequence of variable metric matrices also converges. The limit of this sequence is identified in the case whenF(x) is a strictly convex quadratic function.

## 32 Citations

Convergence of quasi-Newton matrices generated by the symmetric rank one update

- Mathematics, Computer ScienceMath. Program.
- 1991

Conditions under which these approximations can be proved to converge globally to the true Hessian matrix are given, in the case where the Symmetric Rank One update formula is used.

The global convergence of partitioned BFGS on problems with convex decompositions and Lipschitzian gradients

- MathematicsMath. Program.
- 1991

The main purpose of this paper is the extension of Powell's (1976) global convergence result to the partitioned BFGS method introduced by Griewank and Toint (1982), and a damping of the BFGS update that becomes inactive if the problem turns out to be regular nearx*.

The convergence of matrices generated by rank-2 methods from the restricted β-class of Broyden

- Mathematics
- 1984

SummaryIt is shown that the matricesBk generated by any method from the restricted β-class of Broyden converge, if the method is applied to the unconstrained minimization of a functionf∈C2(Rn) with…

Rates of convergence for secant methods on nonlinear problems in hilbert space

- Mathematics
- 1986

The numerical performance of iterative methods applied to discretized operator equations may depend strongly on their theoretical rate of convergence on the underlying problem g(x)=0 in Hilbert…

Solving reachability problems by a scalable constrained optimization method

- Computer ScienceOptimization and Engineering
- 2019

This paper investigates the problem of finding an evolution of a dynamical system that originates and terminates in given sets of states and finds a scalable approach for solving it.

A Theoretical and Experimental Study of the Symmetric Rank-One Update

- Computer ScienceSIAM J. Optim.
- 1993

A new analysis is presented that shows that the SRi method with a line search is $( n + 1)$-step q-superlinearly convergent without the assumption of linearly independent iterates.

Sequential quadratic programming with indefinite Hessian approximations for nonlinear optimum experimental design for parameter estimation in differential–algebraic equations

- Computer Science
- 2015

Algorithms for the numerical solution of problems from nonlinear optimum experimental design (OED) for parameter estimation in differential–algebraic equations and a filter line search globalization strategy that accepts indefinite Hessians based on a new criterion derived from the proof of global convergence are developed.

Convergence properties of the Broyden-like method for mixed linear-nonlinear systems of equations

- MathematicsNumer. Algorithms
- 2021

This is the first time that convergence of the Broyden-like matrices is proven for n > 1, albeit for a special case only and the subspace property of the iterates belongs to an affine subspace is used.

On the convergence of Broyden's method and some accelerated schemes for singular problems

- MathematicsArXiv
- 2021

It is shown that the use of a preceding Newton–like step ensures convergence for starting points in a starlike domain with density 1, and it is established that the matrix updates of Broyden’s method converge q-linearly with the same asymptotic factor as the iterates.

Greedy and Random Broyden's Methods with Explicit Superlinear Convergence Rates in Nonlinear Equations

- Computer Science, MathematicsArXiv
- 2021

This work proposes the greedy and random Broyden’s method for solving nonlinear equations, and establishes explicit (local) superlinear convergence rates of both methods if the initial point and approximate Jacobian are close enough to a solution and corresponding Jacobian.

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