28 research outputs found
Characterizations of Super-regularity and its Variants
Convergence of projection-based methods for nonconvex set feasibility
problems has been established for sets with ever weaker regularity assumptions.
What has not kept pace with these developments is analogous results for
convergence of optimization problems with correspondingly weak assumptions on
the value functions. Indeed, one of the earliest classes of nonconvex sets for
which convergence results were obtainable, the class of so-called super-regular
sets introduced by Lewis, Luke and Malick (2009), has no functional
counterpart. In this work, we amend this gap in the theory by establishing the
equivalence between a property slightly stronger than super-regularity, which
we call Clarke super-regularity, and subsmootheness of sets as introduced by
Aussel, Daniilidis and Thibault (2004). The bridge to functions shows that
approximately convex functions studied by Ngai, Luc and Th\'era (2000) are
those which have Clarke super-regular epigraphs. Further classes of regularity
of functions based on the corresponding regularity of their epigraph are also
discussed.Comment: 15 pages, 2 figure
A Bregman Method for Structure Learning on Sparse Directed Acyclic Graphs
We develop a Bregman proximal gradient method for structure learning on
linear structural causal models. While the problem is non-convex, has high
curvature and is in fact NP-hard, Bregman gradient methods allow us to
neutralize at least part of the impact of curvature by measuring smoothness
against a highly nonlinear kernel. This allows the method to make longer steps
and significantly improves convergence. Each iteration requires solving a
Bregman proximal step which is convex and efficiently solvable for our
particular choice of kernel. We test our method on various synthetic and real
data sets
Bregman Proximal Gradient Algorithm with Extrapolation for a class of Nonconvex Nonsmooth Minimization Problems
In this paper, we consider an accelerated method for solving nonconvex and
nonsmooth minimization problems. We propose a Bregman Proximal Gradient
algorithm with extrapolation(BPGe). This algorithm extends and accelerates the
Bregman Proximal Gradient algorithm (BPG), which circumvents the restrictive
global Lipschitz gradient continuity assumption needed in Proximal Gradient
algorithms (PG). The BPGe algorithm has higher generality than the recently
introduced Proximal Gradient algorithm with extrapolation(PGe), and besides,
due to the extrapolation step, BPGe converges faster than BPG algorithm.
Analyzing the convergence, we prove that any limit point of the sequence
generated by BPGe is a stationary point of the problem by choosing parameters
properly. Besides, assuming Kurdyka-{\'L}ojasiewicz property, we prove the
whole sequences generated by BPGe converges to a stationary point. Finally, to
illustrate the potential of the new method BPGe, we apply it to two important
practical problems that arise in many fundamental applications (and that not
satisfy global Lipschitz gradient continuity assumption): Poisson linear
inverse problems and quadratic inverse problems. In the tests the accelerated
BPGe algorithm shows faster convergence results, giving an interesting new
algorithm.Comment: Preprint submitted for publication, February 14, 201
On Inexact Solution of Auxiliary Problems in Tensor Methods for Convex Optimization
In this paper we study the auxiliary problems that appear in -order tensor
methods for unconstrained minimization of convex functions with
-H\"{o}lder continuous th derivatives. This type of auxiliary problems
corresponds to the minimization of a -order regularization of the
th order Taylor approximation of the objective. For the case , we
consider the use of Gradient Methods with Bregman distance. When the
regularization parameter is sufficiently large, we prove that the referred
methods take at most iterations to find
either a suitable approximate stationary point of the tensor model or an
-approximate stationary point of the original objective function