16 research outputs found
Deflation for semismooth equations
Variational inequalities can in general support distinct solutions. In this
paper we study an algorithm for computing distinct solutions of a variational
inequality, without varying the initial guess supplied to the solver. The
central idea is the combination of a semismooth Newton method with a deflation
operator that eliminates known solutions from consideration. Given one root of
a semismooth residual, deflation constructs a new problem for which a
semismooth Newton method will not converge to the known root, even from the
same initial guess. This enables the discovery of other roots. We prove the
effectiveness of the deflation technique under the same assumptions that
guarantee locally superlinear convergence of a semismooth Newton method. We
demonstrate its utility on various finite- and infinite-dimensional examples
drawn from constrained optimization, game theory, economics and solid
mechanics.Comment: 24 pages, 3 figure
Elastic-Net Regularization: Error estimates and Active Set Methods
This paper investigates theoretical properties and efficient numerical
algorithms for the so-called elastic-net regularization originating from
statistics, which enforces simultaneously l^1 and l^2 regularization. The
stability of the minimizer and its consistency are studied, and convergence
rates for both a priori and a posteriori parameter choice rules are
established. Two iterative numerical algorithms of active set type are
proposed, and their convergence properties are discussed. Numerical results are
presented to illustrate the features of the functional and algorithms
On Quasi-Newton Forward--Backward Splitting: Proximal Calculus and Convergence
We introduce a framework for quasi-Newton forward--backward splitting
algorithms (proximal quasi-Newton methods) with a metric induced by diagonal
rank- symmetric positive definite matrices. This special type of
metric allows for a highly efficient evaluation of the proximal mapping. The
key to this efficiency is a general proximal calculus in the new metric. By
using duality, formulas are derived that relate the proximal mapping in a
rank- modified metric to the original metric. We also describe efficient
implementations of the proximity calculation for a large class of functions;
the implementations exploit the piece-wise linear nature of the dual problem.
Then, we apply these results to acceleration of composite convex minimization
problems, which leads to elegant quasi-Newton methods for which we prove
convergence. The algorithm is tested on several numerical examples and compared
to a comprehensive list of alternatives in the literature. Our quasi-Newton
splitting algorithm with the prescribed metric compares favorably against
state-of-the-art. The algorithm has extensive applications including signal
processing, sparse recovery, machine learning and classification to name a few.Comment: arXiv admin note: text overlap with arXiv:1206.115
Duality results and regularization schemes for Prandtl--Reuss perfect plasticity
We consider the time-discretized problem of the quasi-static evolution problem in perfect plasticity posed in a non-reflexive Banach space and we derive an equivalent version in a reflexive Banach space. A primal-dual stabilization scheme is shown to be consistent with the initial problem. As a consequence, not only stresses, but also displacement and strains are shown to converge to a solution of the original problem in a suitable topology. This scheme gives rise to a well-defined Fenchel dual problem which is a modification of the usual stress problem in perfect plasticity. The dual problem has a simpler structure and turns out to be well-suited for numerical purposes. For the corresponding subproblems an efficient algorithmic approach in the infinite-dimensional setting based on the semismooth Newton method is proposed