2,101 research outputs found
New convergence results for the scaled gradient projection method
The aim of this paper is to deepen the convergence analysis of the scaled
gradient projection (SGP) method, proposed by Bonettini et al. in a recent
paper for constrained smooth optimization. The main feature of SGP is the
presence of a variable scaling matrix multiplying the gradient, which may
change at each iteration. In the last few years, an extensive numerical
experimentation showed that SGP equipped with a suitable choice of the scaling
matrix is a very effective tool for solving large scale variational problems
arising in image and signal processing. In spite of the very reliable numerical
results observed, only a weak, though very general, convergence theorem is
provided, establishing that any limit point of the sequence generated by SGP is
stationary. Here, under the only assumption that the objective function is
convex and that a solution exists, we prove that the sequence generated by SGP
converges to a minimum point, if the scaling matrices sequence satisfies a
simple and implementable condition. Moreover, assuming that the gradient of the
objective function is Lipschitz continuous, we are also able to prove the
O(1/k) convergence rate with respect to the objective function values. Finally,
we present the results of a numerical experience on some relevant image
restoration problems, showing that the proposed scaling matrix selection rule
performs well also from the computational point of view
LSOS: Line-search Second-Order Stochastic optimization methods for nonconvex finite sums
We develop a line-search second-order algorithmic framework for minimizing
finite sums. We do not make any convexity assumptions, but require the terms of
the sum to be continuously differentiable and have Lipschitz-continuous
gradients. The methods fitting into this framework combine line searches and
suitably decaying step lengths. A key issue is a two-step sampling at each
iteration, which allows us to control the error present in the line-search
procedure. Stationarity of limit points is proved in the almost-sure sense,
while almost-sure convergence of the sequence of approximations to the solution
holds with the additional hypothesis that the functions are strongly convex.
Numerical experiments, including comparisons with state-of-the art stochastic
optimization methods, show the efficiency of our approach.Comment: 22 pages, 4 figure
Limited-memory scaled gradient projection methods for real-time image deconvolution in microscopy
Gradient projection methods have given rise to effective tools for image
deconvolution in several relevant areas, such as microscopy, medical imaging
and astronomy. Due to the large scale of the optimization problems arising
in nowadays imaging applications and to the growing request of real-time
reconstructions, an interesting challenge to be faced consists in designing
new acceleration techniques for the gradient schemes, able to preserve the
simplicity and low computational cost of each iteration. In this work we
propose an acceleration strategy for a state of the art scaled gradient
projection method for image deconvolution in microscopy. The acceleration
idea is derived by adapting a step-length selection rule, recently
introduced for limited-memory steepest descent methods in unconstrained
optimization, to the special constrained optimization framework arising in
image reconstruction. We describe how important issues related to the
generalization of the step-length rule to the imaging optimization problem
have been faced and we evaluate the improvements due to the acceleration
strategy by numerical experiments on large-scale image deconvolution problems
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