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

Stoer, Josef

doi:10.1007/bf01389753

Cited by 16 publications

(7 citation statements)

References 14 publications

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“…For other quasi-Newton updates, convergence results are available. We are aware of results for the SR1 update [5,11,30], for the Powell-symmetric-Broyden update [26], for the DFP and the BFGS update [13], and for the convex Broyden class excluding the DFP update [29].…”

Section: Introductionmentioning

confidence: 99%

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

Mannel

2021

Numer Algor

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We consider the Broyden-like method for a nonlinear mapping $F:\mathbb {R}^{n}\rightarrow \mathbb {R}^{n}$ F : ℝ n → ℝ n that has some affine component functions, using an initial matrix B0 that agrees with the Jacobian of F in the rows that correspond to affine components of F. We show that in this setting, the iterates belong to an affine subspace and can be viewed as outcome of the Broyden-like method applied to a lower-dimensional mapping $G:\mathbb {R}^{d}\rightarrow \mathbb {R}^{d}$ G : ℝ d → ℝ d , where d is the dimension of the affine subspace. We use this subspace property to make some small contributions to the decades-old question of whether the Broyden-like matrices converge: First, we observe that the only available result concerning this question cannot be applied if the iterates belong to a subspace because the required uniform linear independence does not hold. By generalizing the notion of uniform linear independence to subspaces, we can extend the available result to this setting. Second, we infer from the extended result that if at most one component of F is nonlinear while the others are affine and the associated n − 1 rows of the Jacobian of F agree with those of B0, then the Broyden-like matrices converge if the iterates converge; this holds whether the Jacobian at the root is invertible or not. In particular, this is the first time that convergence of the Broyden-like matrices is proven for n > 1, albeit for a special case only. Third, under the additional assumption that the Broyden-like method turns into Broyden’s method after a finite number of iterations, we prove that the convergence order of iterates and matrix updates is bounded from below by $\frac {\sqrt {5}+1}{2}$ 5 + 1 2 if the Jacobian at the root is invertible. If the nonlinear component of F is actually affine, we show finite convergence. We provide high-precision numerical experiments to confirm the results.

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Section: Introductionmentioning

confidence: 99%

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

Mannel

2021

Numer Algor

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“…Powell shows in [Pow70] that if ULI is satisfied, then the Powell-symmetric-Broyden matrices converge to the true Hessian and he proposes algorithmic modifications to ensure that ULI holds. In [GP83] it is shown that the DFP and BFGS matrices converge under very general assumptions and in [Sto84] these results are extended to the convex Broyden class excluding DFP. Notably, ULI is not used in [GP83,Sto84] and convergence to limits other than the true Hessian is possible and actually observed.…”

Section: Related Workmentioning

confidence: 99%

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

Mannel¹

2021

Preprint

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We consider Broyden's method and some accelerated schemes for nonlinear equations having a strongly regular singularity of first order with a one-dimensional nullspace. Our two main results are as follows. First, we show that the use of a preceding Newton-like step ensures convergence for starting points in a starlike domain with density 1. This extends the domain of convergence of these methods significantly. Second, we establish that the matrix updates of Broyden's method converge q-linearly with the same asymptotic factor as the iterates. This contributes to the long-standing question whether the Broyden matrices converge by showing that this is indeed the case for the setting at hand. Furthermore, we prove that the Broyden directions violate uniform linear independence, which implies that existing results for convergence of the Broyden matrices cannot be applied. Numerical experiments of high precision confirm the enlarged domain of convergence, the q-linear convergence of the matrix updates, and the lack of uniform linear independence. In addition, they suggest that these results can be extended to singularities of higher order and that Broyden's method can converge r-linearly without converging q-linearly. The underlying code is freely available.

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“…The perturbation argument showing conditions under which (QP) has the same active set as (NLP) is derived from the work of Robinson (1974) who proves a general perturbation result. For a discussion of the convergence of matrix updates see Boggs and Tolle (1994), Ge and Powell (1983) and Stoer (1984).…”

Section: Notes and Referencesmentioning

confidence: 99%

Sequential Quadratic Programming

1995

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Since its popularization in the late 1970s, Sequential Quadratic Programming (SQP) has arguably become the most successful method for solving nonlinearly constrained optimization problems. As with most optimization methods, SQP is not a single algorithm, but rather a conceptual method from which numerous specific algorithms have evolved. Backed by a solid theoretical and computational foundation, both commercial and public-domain SQP algorithms have been developed and used to solve a remarkably large set of important practical problems. Recently large-scale versions have been devised and tested with promising results.

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

Cited by 16 publications

References 14 publications

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

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

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

Sequential Quadratic Programming

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