7,618 research outputs found
Newton-Raphson Consensus for Distributed Convex Optimization
We address the problem of distributed uncon- strained convex optimization
under separability assumptions, i.e., the framework where each agent of a
network is endowed with a local private multidimensional convex cost, is
subject to communication constraints, and wants to collaborate to compute the
minimizer of the sum of the local costs. We propose a design methodology that
combines average consensus algorithms and separation of time-scales ideas. This
strategy is proved, under suitable hypotheses, to be globally convergent to the
true minimizer. Intuitively, the procedure lets the agents distributedly
compute and sequentially update an approximated Newton- Raphson direction by
means of suitable average consensus ratios. We show with numerical simulations
that the speed of convergence of this strategy is comparable with alternative
optimization strategies such as the Alternating Direction Method of
Multipliers. Finally, we propose some alternative strategies which trade-off
communication and computational requirements with convergence speed.Comment: 18 pages, preprint with proof
Optimization Methods for Inverse Problems
Optimization plays an important role in solving many inverse problems.
Indeed, the task of inversion often either involves or is fully cast as a
solution of an optimization problem. In this light, the mere non-linear,
non-convex, and large-scale nature of many of these inversions gives rise to
some very challenging optimization problems. The inverse problem community has
long been developing various techniques for solving such optimization tasks.
However, other, seemingly disjoint communities, such as that of machine
learning, have developed, almost in parallel, interesting alternative methods
which might have stayed under the radar of the inverse problem community. In
this survey, we aim to change that. In doing so, we first discuss current
state-of-the-art optimization methods widely used in inverse problems. We then
survey recent related advances in addressing similar challenges in problems
faced by the machine learning community, and discuss their potential advantages
for solving inverse problems. By highlighting the similarities among the
optimization challenges faced by the inverse problem and the machine learning
communities, we hope that this survey can serve as a bridge in bringing
together these two communities and encourage cross fertilization of ideas.Comment: 13 page
The LBFGS Quasi-Newtonian Method for Molecular Modeling Prion AGAAAAGA Amyloid Fibrils
Experimental X-ray crystallography, NMR (Nuclear Magnetic Resonance)
spectroscopy, dual polarization interferometry, etc are indeed very powerful
tools to determine the 3-Dimensional structure of a protein (including the
membrane protein); theoretical mathematical and physical computational
approaches can also allow us to obtain a description of the protein 3D
structure at a submicroscopic level for some unstable, noncrystalline and
insoluble proteins. X-ray crystallography finds the X-ray final structure of a
protein, which usually need refinements using theoretical protocols in order to
produce a better structure. This means theoretical methods are also important
in determinations of protein structures. Optimization is always needed in the
computer-aided drug design, structure-based drug design, molecular dynamics,
and quantum and molecular mechanics. This paper introduces some optimization
algorithms used in these research fields and presents a new theoretical
computational method - an improved LBFGS Quasi-Newtonian mathematical
optimization method - to produce 3D structures of Prion AGAAAAGA amyloid
fibrils (which are unstable, noncrystalline and insoluble), from the potential
energy minimization point of view. Because the NMR or X-ray structure of the
hydrophobic region AGAAAAGA of prion proteins has not yet been determined, the
model constructed by this paper can be used as a reference for experimental
studies on this region, and may be useful in furthering the goals of medicinal
chemistry in this field
Phase retrieval and saddle-point optimization
Iterative algorithms with feedback are amongst the most powerful and
versatile optimization methods for phase retrieval. Among these, the hybrid
input-output algorithm has demonstrated practical solutions to giga-element
nonlinear phase retrieval problems, escaping local minima and producing images
at resolutions beyond the capabilities of lens-based optical methods. Here, the
input-output iteration is improved by a lower dimensional subspace saddle-point
optimization.Comment: 8 pages, 4 figures, revte
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