195 research outputs found
Unconstrained Optimization Methods: Conjugate Gradient Methods and Trust-Region Methods
Here, we consider two important classes of unconstrained optimization methods: conjugate gradient methods and trust region methods. These two classes of methods are very interesting; it seems that they are never out of date. First, we consider conjugate gradient methods. We also illustrate the practical behavior of some conjugate gradient methods. Then, we study trust region methods. Considering these two classes of methods, we analyze some recent results
Scaled Projected-Directions Methods with Application to Transmission Tomography
Statistical image reconstruction in X-Ray computed tomography yields
large-scale regularized linear least-squares problems with nonnegativity
bounds, where the memory footprint of the operator is a concern. Discretizing
images in cylindrical coordinates results in significant memory savings, and
allows parallel operator-vector products without on-the-fly computation of the
operator, without necessarily decreasing image quality. However, it
deteriorates the conditioning of the operator. We improve the Hessian
conditioning by way of a block-circulant scaling operator and we propose a
strategy to handle nondiagonal scaling in the context of projected-directions
methods for bound-constrained problems. We describe our implementation of the
scaling strategy using two algorithms: TRON, a trust-region method with exact
second derivatives, and L-BFGS-B, a linesearch method with a limited-memory
quasi-Newton Hessian approximation. We compare our approach with one where a
change of variable is made in the problem. On two reconstruction problems, our
approach converges faster than the change of variable approach, and achieves
much tighter accuracy in terms of optimality residual than a first-order
method.Comment: 19 pages, 7 figure
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
Extension of Modified Polak-Ribière-Polyak Conjugate Gradient Method to Linear Equality Constraints Minimization Problems
Combining the Rosen gradient projection method with the two-term Polak-Ribière-Polyak (PRP) conjugate gradient method, we propose a two-term Polak-Ribière-Polyak (PRP) conjugate gradient projection method for solving linear equality constraints optimization problems. The proposed method possesses some attractive properties: (1) search direction generated by the proposed method is a feasible descent direction; consequently the generated iterates are feasible points; (2) the sequences of function are decreasing. Under some mild conditions, we show that it is globally convergent with Armijio-type line search. Preliminary numerical results show that the proposed method is promising
Optimization with Sparsity-Inducing Penalties
Sparse estimation methods are aimed at using or obtaining parsimonious
representations of data or models. They were first dedicated to linear variable
selection but numerous extensions have now emerged such as structured sparsity
or kernel selection. It turns out that many of the related estimation problems
can be cast as convex optimization problems by regularizing the empirical risk
with appropriate non-smooth norms. The goal of this paper is to present from a
general perspective optimization tools and techniques dedicated to such
sparsity-inducing penalties. We cover proximal methods, block-coordinate
descent, reweighted -penalized techniques, working-set and homotopy
methods, as well as non-convex formulations and extensions, and provide an
extensive set of experiments to compare various algorithms from a computational
point of view
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