4,696 research outputs found
Local convergence of quasi-Newton methods under metric regularity
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.A. Belyakov was supported by the Austrian Science Foundation (FWF) under grant No P 24125-N13. A.L. Dontchev was supported by NSF Grant DMS 1008341 through the University of Michigan. M. López was supported by MINECO of Spain, Grant MTM2011-29064-C03-02
Strong Metric (Sub)regularity of KKT Mappings for Piecewise Linear-Quadratic Convex-Composite Optimization
This work concerns the local convergence theory of Newton and quasi-Newton
methods for convex-composite optimization: minimize f(x):=h(c(x)), where h is
an infinite-valued proper convex function and c is C^2-smooth. We focus on the
case where h is infinite-valued piecewise linear-quadratic and convex. Such
problems include nonlinear programming, mini-max optimization, estimation of
nonlinear dynamics with non-Gaussian noise as well as many modern approaches to
large-scale data analysis and machine learning. Our approach embeds the
optimality conditions for convex-composite optimization problems into a
generalized equation. We establish conditions for strong metric subregularity
and strong metric regularity of the corresponding set-valued mappings. This
allows us to extend classical convergence of Newton and quasi-Newton methods to
the broader class of non-finite valued piecewise linear-quadratic
convex-composite optimization problems. In particular we establish local
quadratic convergence of the Newton method under conditions that parallel those
in nonlinear programming when h is non-finite valued piecewise linear
Gravitational Collapse in Einstein dilaton Gauss-Bonnet Gravity
We present results from a numerical study of spherical gravitational collapse
in shift symmetric Einstein dilaton Gauss-Bonnet (EdGB) gravity. This modified
gravity theory has a single coupling parameter that when zero reduces to
general relativity (GR) minimally coupled to a massless scalar field. We first
show results from the weak EdGB coupling limit, where we obtain solutions that
smoothly approach those of the Einstein-Klein-Gordon system of GR. Here, in the
strong field case, though our code does not utilize horizon penetrating
coordinates, we nevertheless find tentative evidence that approaching black
hole formation the EdGB modifications cause the growth of scalar field "hair",
consistent with known static black hole solutions in EdGB gravity. For the
strong EdGB coupling regime, in a companion paper we first showed results that
even in the weak field (i.e. far from black hole formation), the EdGB equations
are of mixed type: evolution of the initially hyperbolic system of partial
differential equations lead to formation of a region where their character
changes to elliptic. Here, we present more details about this regime. In
particular, we show that an effective energy density based on the Misner-Sharp
mass is negative near these elliptic regions, and similarly the null
convergence condition is violated then.Comment: 35 pages, 11 figures, edited to resemble journal versio
A Bregman forward-backward linesearch algorithm for nonconvex composite optimization: superlinear convergence to nonisolated local minima
We introduce Bella, a locally superlinearly convergent Bregman forward
backward splitting method for minimizing the sum of two nonconvex functions,
one of which satisfying a relative smoothness condition and the other one
possibly nonsmooth. A key tool of our methodology is the Bregman
forward-backward envelope (BFBE), an exact and continuous penalty function with
favorable first- and second-order properties, and enjoying a nonlinear error
bound when the objective function satisfies a Lojasiewicz-type property. The
proposed algorithm is of linesearch type over the BFBE along candidate update
directions, and converges subsequentially to stationary points, globally under
a KL condition, and owing to the given nonlinear error bound can attain
superlinear convergence rates even when the limit point is a nonisolated
minimum, provided the directions are suitably selected
- …