88 research outputs found
String-Averaging Projected Subgradient Methods for Constrained Minimization
We consider constrained minimization problems and propose to replace the
projection onto the entire feasible region, required in the Projected
Subgradient Method (PSM), by projections onto the individual sets whose
intersection forms the entire feasible region. Specifically, we propose to
perform such projections onto the individual sets in an algorithmic regime of a
feasibility-seeking iterative projection method. For this purpose we use the
recently developed family of Dynamic String-Averaging Projection (DSAP) methods
wherein iteration-index-dependent variable strings and variable weights are
permitted. This gives rise to an algorithmic scheme that generalizes, from the
algorithmic structural point of view, earlier work of Helou Neto and De Pierro,
of Nedi\'c, of Nurminski, and of Ram et al.Comment: Optimization Methods and Software, accepted for publicatio
Convergence of generic infinite products of affine operators
We establish several results concerning the asymptotic behavior of random infinite products of generic sequences of affine uniformly continuous operators on bounded closed convex subsets of a Banach space. In addition to weak ergodic theorems we also obtain convergence to a unique common fixed point and more generally, to an affine retraction
Convergence and Perturbation Resilience of Dynamic String-Averaging Projection Methods
We consider the convex feasibility problem (CFP) in Hilbert space and
concentrate on the study of string-averaging projection (SAP) methods for the
CFP, analyzing their convergence and their perturbation resilience. In the
past, SAP methods were formulated with a single predetermined set of strings
and a single predetermined set of weights. Here we extend the scope of the
family of SAP methods to allow iteration-index-dependent variable strings and
weights and term such methods dynamic string-averaging projection (DSAP)
methods. The bounded perturbation resilience of DSAP methods is relevant and
important for their possible use in the framework of the recently developed
superiorization heuristic methodology for constrained minimization problems.Comment: Computational Optimization and Applications, accepted for publicatio
On the Relation Between Turnpike Properties for Finite and Infinite Horizon Optimal Control Problems
Greedy optimal control for elliptic problems and its application to turnpike problems
This is a post-peer-review, pre-copyedit version of an article published in Numerische Mathematik. The final authenticated version is available online at: https://doi.org/10.1007/s00211-018-1005-zWe adapt and apply greedy methods to approximate in an efficient way the optimal controls for parameterized elliptic control problems. Our results yield an optimal approximation procedure that, in particular, performs better than simply sampling the parameter-space to compute controls for each parameter value. The same method can be adapted for parabolic control problems, but this leads to greedy selections of the realizations of the parameters that depend on the initial datum under consideration. The turnpike property (which ensures that parabolic optimal control problems behave nearly in a static manner when the control horizon is long enough) allows using the elliptic greedy choice of the parameters in the parabolic setting too. We present various numerical experiments and an extensive discussion of the efficiency of our methodology for parabolic control and indicate a number of open problems arising when analyzing the convergence of the proposed algorithmsThis project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 694126-DyCon). Part of this research was done while the second author visited DeustoTech and Univesity of Deusto with the support of the DyCon project. The second author was also partially supported by Croatian Science Foundation under ConDyS Project, IP-2016-06-2468. The work of the third author was partially supported by the Grants MTM2014-52347, MTM2017-92996 of MINECO (Spain) and ICON of the French AN
Arrows of Time and Chaotic Properties of the Cosmic Background Radiation
We advance a new viewpoint on the connection between the the thermodynamical
and cosmological arrows of time, which can be traced via the properties of
Cosmic Microwave Background (CMB) radiation. We show that in the
Friedmann-Robertson-Walker Universe with negative curvature there is a
necessary ingredient for the existence of the thermodynamical arrow of time. It
is based on the dynamical instability of motion along null geodesics in a
hyperbolic space. Together with special (de-correlated) initial conditions,
this mechanism is sufficient for the thermodynamical arrow, whereas the special
initial conditions alone are able to generate only a pre-arrow of time. Since
the negatively curved space will expand forever, this provides a direct
connection between the thermodynamical and cosmological arrows of time. The
structural stability of the geodesic flows on hyperbolic spaces and hence the
robustness of the proposed mechanism is especially stressed. We then point out
that the main relations of equilibrium statistical thermodynamics (including
the second law) do not necessarily depend on any arrow of time. Finally we
formulate a {\it curvature anthropic principle}, which stipulates the negative
curvature as a necessary condition for the time asymmetric Universe with an
observer. CMB has to carry the signature of this principle as well
Magnetoplasmon emission versus Landau-level scattering in resonant tunneling through double-barrier structures
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