1,080 research outputs found
Global convergence of splitting methods for nonconvex composite optimization
We consider the problem of minimizing the sum of a smooth function with a
bounded Hessian, and a nonsmooth function. We assume that the latter function
is a composition of a proper closed function and a surjective linear map
, with the proximal mappings of , , simple to
compute. This problem is nonconvex in general and encompasses many important
applications in engineering and machine learning. In this paper, we examined
two types of splitting methods for solving this nonconvex optimization problem:
alternating direction method of multipliers and proximal gradient algorithm.
For the direct adaptation of the alternating direction method of multipliers,
we show that, if the penalty parameter is chosen sufficiently large and the
sequence generated has a cluster point, then it gives a stationary point of the
nonconvex problem. We also establish convergence of the whole sequence under an
additional assumption that the functions and are semi-algebraic.
Furthermore, we give simple sufficient conditions to guarantee boundedness of
the sequence generated. These conditions can be satisfied for a wide range of
applications including the least squares problem with the
regularization. Finally, when is the identity so that the proximal
gradient algorithm can be efficiently applied, we show that any cluster point
is stationary under a slightly more flexible constant step-size rule than what
is known in the literature for a nonconvex .Comment: To appear in SIOP
Blind Ptychographic Phase Retrieval via Convergent Alternating Direction Method of Multipliers
Ptychography has risen as a reference X-ray imaging technique: it achieves
resolutions of one billionth of a meter, macroscopic field of view, or the
capability to retrieve chemical or magnetic contrast, among other features. A
ptychographyic reconstruction is normally formulated as a blind phase retrieval
problem, where both the image (sample) and the probe (illumination) have to be
recovered from phaseless measured data. In this article we address a nonlinear
least squares model for the blind ptychography problem with constraints on the
image and the probe by maximum likelihood estimation of the Poisson noise
model. We formulate a variant model that incorporates the information of
phaseless measurements of the probe to eliminate possible artifacts. Next, we
propose a generalized alternating direction method of multipliers designed for
the proposed nonconvex models with convergence guarantee under mild conditions,
where their subproblems can be solved by fast element-wise operations.
Numerically, the proposed algorithm outperforms state-of-the-art algorithms in
both speed and image quality.Comment: 23 page
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