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Concept

Point col

Résumé

En mathématiques, un point col ou point-selle () d'une fonction f définie sur un produit cartésien X × Y de deux ensembles X et Y est un point (\bar{x},\bar{y})\in X\times Y tel que :

y\mapsto f(\bar{x},y) atteint un maximum en \bar{y} sur Y ;
et x\mapsto f(x,\bar{y}) atteint un minimum en \bar{x} sur X. Certains auteurs inversent les maximum et minimum (f(\bar{x},\cdot) a un minimum en \bar{y} et f(\cdot,\bar{y}) a un maximum en \bar{x}), mais cela ne modifie pas qualitativement les résultats (on peut revenir au cas présent par un changement de variables).

Le terme point-selle fait référence à la forme de selle de cheval que prend le graphe de la fonction lorsque X et Y sont des intervalles de \R. Le terme de point col, renvoie, quant à lui, à l'image du col

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Escaping from saddle points on Riemannian manifolds

Nicolas Henri Bernard Flammarion, Yue Sun

We consider minimizing a nonconvex, smooth function f on a Riemannian manifold M. We show that a perturbed version of Riemannian gradient descent algorithm converges to a second-order stationary point (and hence is able to escape saddle points on the manifold). The rate of convergence depends as 1/epsilon(2) on the accuracy c, which matches a rate known only for unconstrained smooth minimization. The convergence rate depends polylogarithmically on the manifold dimension d, hence is almost dimension -free. The rate also has a polynomial dependence on the parameters describing the curvature of the manifold and the smoothness of the function. While the unconstrained problem (Euclidean setting) is well -studied, our result is the first to prove such a rate for nonconvex, manifold-constrained problems.

2019

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We consider the problem of finding a saddle point for the convex-concave objective

\min_x \max_y f(x) + \langle Ax, y\rangle - g^*(y)

, where

f

is a convex function with locally Lipschitz gradient and

g

is convex and possibly non-smooth. We propose an adaptive version of the Condat-Vũ algorithm, which alternates between primal gradient steps and dual proximal steps. The method achieves stepsize adaptivity through a simple rule involving

\|A\|

and the norm of recently computed gradients of

f

. Under standard assumptions, we prove an

\mathcal{O}(k^{-1})

ergodic convergence rate. Furthermore, when

f

is also locally strongly convex and

A

has full row rank we show that our method converges with a linear rate. Numerical experiments are provided for illustrating the practical performance of the algorithm.

2021

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Minimum energy path and atomistic mechanism of the elementary step in oxygen diffusion in silicon: A density-functional study

Jan Felix Binder, Alfredo Pasquarello

Using a density-functional scheme, we study the migration of a single O atom in a (110) plane between two adjacent bond-center sites in bulk Si. The minimum energy migration path is found through the nudged elastic band method within a generalized gradient approximation for the electronic structure. The energy barrier is then also evaluated within a hybrid functional scheme. We achieve for the transition barrier a best estimate of 2.3 eV in the generalized gradient approximation and of 2.7 eV in the hybrid functional scheme, both in fair agreement with the commonly accepted experimental value of 2.53 eV. The transition is characterized by a saddle point which does not occur at the midpoint between the two bond-center sites and by a pattern of displacements extending up to the second nearest-neighbor Si atoms. The atomistic mechanism of oxygen migration is analyzed from three complementary viewpoints involving the evolution of the structure, the Wannier centers, and the single-particle energies and wave functions. The diffusion process can be separated into two distinct parts. In one part, the exchange of the Si atoms in the first-neighbor shell of the diffusing O atom occurs through the formation of a threefold coordinated O center and an overcoordinated Si atom. In the other part, the Si-Si bond flips its position through the creation of occupied and unoccupied Si dangling bonds which give rise to states in the band gap.

American Physical Society2014

Chargement

We consider the problem of finding a saddle point for the convex-concave objective

\min_x \max_y f(x) + \langle Ax, y\rangle - g^*(y)

, where

f

is a convex function with locally Lipschitz gradient and

g

\|A\|

and the norm of recently computed gradients of

f

. Under standard assumptions, we prove an

\mathcal{O}(k^{-1})

ergodic convergence rate. Furthermore, when

f

is also locally strongly convex and

A

has full row rank we show that our method converges with a linear rate. Numerical experiments are provided for illustrating the practical performance of the algorithm.

2021