On a semismooth* Newton method for solving generalized equations

H. Gfrerer; J.V. Outrata

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On a semismooth* Newton method for solving generalized equations

H. Gfrerer, J.V. Outrata

Optimization and Control · Submitted 19 Apr 2019

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The authors' main claim The method is based on the new notion of semismoothness${}^*$ which, together with a suitable regularity condition, ensure the local superlinear convergence.

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In the paper, a Newton-type method for the solution of generalized equations (GEs) is derived, where the linearization concerns both the single-valued and the multi-valued part of the considered GE. The method is based on the new notion of semismoothness${}^*$ which, together with a suitable regularity condition, ensure the local superlinear convergence. An implementable version of the new method is derived for a class of GEs, frequently arising in optimization and equilibrium models.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In the paper, a Newton-type method for the solution of generalized equations (GEs) is derived, where the linearization concerns both the single-valued and the multi-valued part of the considered GE.	contribution 0.73	0.94	0.14	asserted 1.00
s1	The method is based on the new notion of semismoothness${}^*$ which, together with a suitable regularity condition, ensure the local superlinear convergence.	contribution 0.44	0.85	0.46	asserted 1.00
s2	An implementable version of the new method is derived for a class of GEs, frequently arising in optimization and equilibrium models.	contribution 0.84	0.94	0.14	asserted 1.00

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Original abstract

In the paper, a Newton-type method for the solution of generalized equations (GEs) is derived, where the linearization concerns both the single-valued and the multi-valued part of the considered GE. The method is based on the new notion of semismoothness${}^*$ which, together with a suitable regularity condition, ensure the local superlinear convergence. An implementable version of the new method is derived for a class of GEs, frequently arising in optimization and equilibrium models.

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5 arXiv papers cite this in 27 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 19 sentences without a confident label

Uses its method or tool 2 papers

As mentioned above, in this paper we will investigate the numerical solution of GE ([ref]) via the semismooth* Newton method developed in [ref] .

On the application of the semismooth* Newton method to variational inequalities of the second kind · Introduction · 0.98
The solutions can be ordered in vectors $v̂^i = (v̂^i_12, v̂^i_3, v̂^i_4) ∈ℝ^4$ and all together in a vector [equation] Thereafter we compute the outcome of the approximation step via [equation] Clearly $(x̂, d̂, ŷ) ∈ℱ$ and, using the theory [ref] , it is possible to show that condition [ref] is fulfilled.

On the solution of contact problems with Tresca friction by the semismooth* Newton method · Implementation of the method · 0.84

Cites it as support 1 paper

It follows from [ref] that the mapping $T^-1$ is semismooth$^*$ at $(,)$ and so is $T$ at $(,)$.

Generalized Newton Algorithms for Tilt-Stable Minimizers in Nonsmooth Optimization · Newton Algorithms for Prox-Regular Functions · 0.97

Cites it as background 2 papers

The theoretical background of the Newton method has been established in [ref] .

On the application of the semismooth* Newton method to variational inequalities of the second kind · Conclusion · 0.99
Recall from [ref] that a set-valued mapping $F^n^m$ is semismooth$^*$ at $(,)∈F$ if whenever $(u,v)∈^n×^m$ we have [equation] where $D^*F((,);(u,v))$ is the directional coderivative of $F$ at $(,)∈F$ in the direction $(u,v)∈^n×^m$ defined by [equation] via the the directional normal cone [equation] to $F$ at $(,)$ in the direction $(u,v)$.

Generalized Newton Algorithms for Tilt-Stable Minimizers in Nonsmooth Optimization · Major Definitions and Preliminaries · 0.81

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