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Local convergence of quasi-Newton methods under metric regularity Aragón Artacho, Francisco Javier ; ; et al in Computational Optimization and Applications (2014), 58(1), 225-247 We consider quasi-Newton methods for generalized equations in Banach spaces under metric regularity and give a sufficient condition for q-linear convergence. Then we show that the well-known Broyden ... [more ▼] We consider quasi-Newton methods for generalized equations in Banach spaces under metric regularity and give a sufficient condition for q-linear convergence. Then we show that the well-known Broyden update satisfies this sufficient condition in Hilbert spaces. We also establish various modes of q-superlinear convergence of the Broyden update under strong metric subregularity, metric regularity and strong metric regularity. In particular, we show that the Broyden update applied to a generalized equation in Hilbert spaces satisfies the Dennis–Moré condition for q-superlinear convergence. Simple numerical examples illustrate the results. [less ▲] Detailed reference viewed: 84 (12 UL)Metric regularity of Newton's iteration Aragón Artacho, Francisco Javier ; ; et al in SIAM Journal on Control & Optimization (2011), 49(2), 339-362 For a version of Newton's method applied to a generalized equation with a parameter, we extend the paradigm of the Lyusternik–Graves theorem to the framework of a mapping acting from the pair “parameter ... [more ▼] For a version of Newton's method applied to a generalized equation with a parameter, we extend the paradigm of the Lyusternik–Graves theorem to the framework of a mapping acting from the pair “parameter-starting point” to the set of corresponding convergent Newton sequences. Under ample parameterization, metric regularity of the mapping associated with convergent Newton sequences becomes equivalent to the metric regularity of the mapping associated with the generalized equation. We also discuss an inexact Newton method and present an application to discretized optimal control. [less ▲] Detailed reference viewed: 64 (7 UL)On the inner and outer norms of sublinear mappings Aragón Artacho, Francisco Javier ; in Set-Valued Analysis (2007), 15(1), 61-65 In this short note we show that the outer norm of a sublinear mapping F, acting between Banach spaces X and Y and with dom F = X, is finite only if F is single-valued. This implies in particular that for ... [more ▼] In this short note we show that the outer norm of a sublinear mapping F, acting between Banach spaces X and Y and with dom F = X, is finite only if F is single-valued. This implies in particular that for a sublinear multivalued mapping the inner and the outer norms cannot be finite simultaneously. [less ▲] Detailed reference viewed: 58 (2 UL) |
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