Convex hull | EPFL Graph Search

In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points. The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its dual problem of intersecting half-spaces, are fundamental problems of computation

Representing groups against all odds

Maxime Gheysens

We investigate how probability tools can be useful to study representations of non-amenable groups. A suitable notion of "probabilistic subgroup" is proposed for locally compact groups, and is valuable to induction of representations. Nonamenable groups admit nonabelian free subgroups in that measure-theoretical sense. Consequences for affine actions and for unitarizability are then drawn. In particular, we obtain a new characterization of amenability via some affine actions on Hilbert spaces. Along the way, various fixed-point properties for groups are studied. We also give a survey of several useful facts about group representations on Banach spaces, continuity of group actions, compactness of convex hulls in locally convex spaces, and measurability pathologies in Banach spaces.

EPFL2017

Application aux équations aux dérivées partielles d'une méthode par point fixe et le problème des deux puits

Sébastien Basterrechea

This thesis consists of two parts. The first part is about a variant of Banach's fixed point theorem and its applications to several partial differential equations (PDE's), abstractly of the form [ \mathcal Lu + \mathcal Q(u) = f.] The main result of this first part asserts that an equation having this form admits a solution if the datum

f

satisfies a certain smallness assumption. This result (we call it \emph{the fixed point method}) is relatively simple to use and can be applied to a large variety of PDE's. The downside is that it guarantees the existence of solutions only for "small" data. The equations we deal with are Jacobian equations, non-linear elliptic PDE's, transport problems and the semi-linear wave equation. The second part of the thesis treats the

\emph{two well problem}

in two dimensions [ \nabla u \in \mathbb{S}_A \cup \mathbb{S}_B\quad \text{almost everywhere in}\ \Omega, ] [ u = u_0\quad \text{on}\ \partial\Omega. ] For the non-degenerate case

\det(A),\det(B) \neq 0

, we show a non-existence result for piecewise regular solutions if

A

and

B

are non-orthogonal. For the degenerate and semi-degenerate cases, we give a characterisation for the rank-one convex hull of

\mathbb{S}_A \cup \mathbb{S}_B

and several existence results for Lipschitz and piecewise affine solutions. Finally, for each case, we construct several explicit non-trivial solutions for well-chosen boundary conditions

u_0

EPFL2015

Finding stationary points on bounded-rank matrices: a geometric hurdle and a smooth remedy

Nicolas Boumal

We consider the problem of provably finding a stationary point of a smooth function to be minimized on the variety of bounded-rank matrices. This turns out to be unexpectedly delicate. We trace the difficulty back to a geometric obstacle: On a nonsmooth set, there may be sequences of points along which standard measures of stationarity tend to zero, but whose limit points are not stationary. We name such events apocalypses, as they can cause optimization algorithms to converge to non-stationary points. We illustrate this explicitly for an existing optimization algorithm on bounded-rank matrices. To provably find stationary points, we modify a trust-region method on a standard smooth parameterization of the variety. The method relies on the known fact that second-order stationary points on the parameter space map to stationary points on the variety. Our geometric observations and proposed algorithm generalize beyond bounded-rank matrices. We give a geometric characterization of apocalypses on general constraint sets, which implies that Clarke-regular sets do not admit apocalypses. Such sets include smooth manifolds, manifolds with boundaries, and convex sets. Our trust-region method supports parameterization by any complete Riemannian manifold.

SPRINGER HEIDELBERG2022