399 research outputs found
A Multi-step Inertial Forward--Backward Splitting Method for Non-convex Optimization
In this paper, we propose a multi-step inertial Forward--Backward splitting
algorithm for minimizing the sum of two non-necessarily convex functions, one
of which is proper lower semi-continuous while the other is differentiable with
a Lipschitz continuous gradient. We first prove global convergence of the
scheme with the help of the Kurdyka-{\L}ojasiewicz property. Then, when the
non-smooth part is also partly smooth relative to a smooth submanifold, we
establish finite identification of the latter and provide sharp local linear
convergence analysis. The proposed method is illustrated on a few problems
arising from statistics and machine learning.Comment: This paper is in company with our recent work on
Forward--Backward-type splitting methods http://arxiv.org/abs/1503.0370
On the convergence of a linesearch based proximal-gradient method for nonconvex optimization
We consider a variable metric linesearch based proximal gradient method for
the minimization of the sum of a smooth, possibly nonconvex function plus a
convex, possibly nonsmooth term. We prove convergence of this iterative
algorithm to a critical point if the objective function satisfies the
Kurdyka-Lojasiewicz property at each point of its domain, under the assumption
that a limit point exists. The proposed method is applied to a wide collection
of image processing problems and our numerical tests show that our algorithm
results to be flexible, robust and competitive when compared to recently
proposed approaches able to address the optimization problems arising in the
considered applications
A proximal minimization algorithm for structured nonconvex and nonsmooth problems
We propose a proximal algorithm for minimizing objective functions consisting
of three summands: the composition of a nonsmooth function with a linear
operator, another nonsmooth function, each of the nonsmooth summands depending
on an independent block variable, and a smooth function which couples the two
block variables. The algorithm is a full splitting method, which means that the
nonsmooth functions are processed via their proximal operators, the smooth
function via gradient steps, and the linear operator via matrix times vector
multiplication. We provide sufficient conditions for the boundedness of the
generated sequence and prove that any cluster point of the latter is a KKT
point of the minimization problem. In the setting of the Kurdyka-\L{}ojasiewicz
property we show global convergence, and derive convergence rates for the
iterates in terms of the \L{}ojasiewicz exponent
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