2,287 research outputs found
Barrier subgradient method
In this paper we develop a new primal-dual subgradient method for nonsmooth convex optimization problems. This scheme is based on a self-concordant barrier for the basic feasible set. It is suitable for finding approximate solutions with certain relative accuracy. We discuss some applications of this technique including fractional covering problem, maximal concurrent flow problem, semidefinite relaxations and nonlinear online optimization.convex optimization, subgradient methods, non-smooth optimization, minimax problems, saddle points, variational inequalities, stochastic optimization, black-box methods, lower complexity bounds.
Mesoscopic superconductivity in ultrasmall metallic grains
A nano-scale metallic grain (nanoparticle) with irregular boundaries in which
the single-particle dynamics are chaotic is a zero-dimensional system described
by the so-called universal Hamiltonian in the limit of a large number of
electrons. The interaction part of this Hamiltonian includes a superconducting
pairing term and a ferromagnetic exchange term. Spin-orbit scattering breaks
spin symmetry and suppresses the exchange interaction term. Of particular
interest is the fluctuation-dominated regime, typical of the smallest grains in
the experiments, in which the bulk pairing gap is comparable to or smaller than
the single-particle mean-level spacing, and the Bardeen-Cooper-Schrieffer (BCS)
mean-field theory of superconductivity is no longer valid. Here we study the
crossover between the BCS and fluctuation-dominated regimes in two limits. In
the absence of spin-orbit scattering, the pairing and exchange interaction
terms compete with each other. We describe the signatures of this competition
in thermodynamic observables, the heat capacity and spin susceptibility. In the
presence of strong spin-orbit scattering, the exchange interaction term can be
ignored. We discuss how the magnetic-field response of discrete energy levels
in such a nanoparticle is affected by pairing correlations. We identify
signatures of pairing correlations in this response, which are detectable even
in the fluctuation-dominated regime.Comment: 9 pages, 5 figures, Proceedings of the Fourth Conference on Nuclei
and Mesoscopic Physics (NMP14
Magnetic response of energy levels of superconducting nanoparticles with spin-orbit scattering
Discrete energy levels of ultrasmall metallic grains are extracted in
single-electron-tunneling-spectroscopy experiments. We study the response of
these energy levels to an external magnetic field in the presence of both
spin-orbit scattering and pairing correlations. In particular, we investigate
-factors and level curvatures that parametrize, respectively, the linear and
quadratic terms in the magnetic-field dependence of the many-particle energy
levels of the grain. Both of these quantities exhibit level-to-level
fluctuations in the presence of spin-orbit scattering. We show that the
distribution of -factors is not affected by the pairing interaction and that
the distribution of level curvatures is sensitive to pairing correlations even
in the smallest grains in which the pairing gap is smaller than the mean
single-particle level spacing. We propose the level curvature in a magnetic
field as a tool to probe pairing correlations in tunneling spectroscopy
experiments.Comment: 13 pages, 5 figure
Minimal Envy and Popular Matchings
We study ex-post fairness in the object allocation problem where objects are
valuable and commonly owned. A matching is fair from individual perspective if
it has only inevitable envy towards agents who received most preferred objects
-- minimal envy matching. A matching is fair from social perspective if it is
supported by majority against any other matching -- popular matching.
Surprisingly, the two perspectives give the same outcome: when a popular
matching exists it is equivalent to a minimal envy matching.
We show the equivalence between global and local popularity: a matching is
popular if and only if there does not exist a group of size up to 3 agents that
decides to exchange their objects by majority, keeping the remaining matching
fixed. We algorithmically show that an arbitrary matching is path-connected to
a popular matching where along the path groups of up to 3 agents exchange their
objects by majority. A market where random groups exchange objects by majority
converges to a popular matching given such matching exists.
When popular matching might not exist we define most popular matching as a
matching that is popular among the largest subset of agents. We show that each
minimal envy matching is a most popular matching and propose a polynomial-time
algorithm to find them
Differentially Private Distributed Optimization
In distributed optimization and iterative consensus literature, a standard
problem is for agents to minimize a function over a subset of Euclidean
space, where the cost function is expressed as a sum . In this paper,
we study the private distributed optimization (PDOP) problem with the
additional requirement that the cost function of the individual agents should
remain differentially private. The adversary attempts to infer information
about the private cost functions from the messages that the agents exchange.
Achieving differential privacy requires that any change of an individual's cost
function only results in unsubstantial changes in the statistics of the
messages. We propose a class of iterative algorithms for solving PDOP, which
achieves differential privacy and convergence to the optimal value. Our
analysis reveals the dependence of the achieved accuracy and the privacy levels
on the the parameters of the algorithm. We observe that to achieve
-differential privacy the accuracy of the algorithm has the order of
Mirror Descent and Convex Optimization Problems With Non-Smooth Inequality Constraints
We consider the problem of minimization of a convex function on a simple set
with convex non-smooth inequality constraint and describe first-order methods
to solve such problems in different situations: smooth or non-smooth objective
function; convex or strongly convex objective and constraint; deterministic or
randomized information about the objective and constraint. We hope that it is
convenient for a reader to have all the methods for different settings in one
place. Described methods are based on Mirror Descent algorithm and switching
subgradient scheme. One of our focus is to propose, for the listed different
settings, a Mirror Descent with adaptive stepsizes and adaptive stopping rule.
This means that neither stepsize nor stopping rule require to know the
Lipschitz constant of the objective or constraint. We also construct Mirror
Descent for problems with objective function, which is not Lipschitz
continuous, e.g. is a quadratic function. Besides that, we address the problem
of recovering the solution of the dual problem
Narrow scope for resolution-limit-free community detection
Detecting communities in large networks has drawn much attention over the
years. While modularity remains one of the more popular methods of community
detection, the so-called resolution limit remains a significant drawback. To
overcome this issue, it was recently suggested that instead of comparing the
network to a random null model, as is done in modularity, it should be compared
to a constant factor. However, it is unclear what is meant exactly by
"resolution-limit-free", that is, not suffering from the resolution limit.
Furthermore, the question remains what other methods could be classified as
resolution-limit-free. In this paper we suggest a rigorous definition and
derive some basic properties of resolution-limit-free methods. More
importantly, we are able to prove exactly which class of community detection
methods are resolution-limit-free. Furthermore, we analyze which methods are
not resolution-limit-free, suggesting there is only a limited scope for
resolution-limit-free community detection methods. Finally, we provide such a
natural formulation, and show it performs superbly
The coexistence of superconductivity and ferromagnetism in nano-scale metallic grains
A nano-scale metallic grain in which the single-particle dynamics are chaotic
is described by the so-called universal Hamiltonian. This Hamiltonian includes
a superconducting pairing term and a ferromagnetic exchange term that compete
with each other: pairing correlations favor minimal ground-state spin, while
the exchange interaction favors maximal spin polarization. Of particular
interest is the fluctuation-dominated regime where the bulk pairing gap is
comparable to or smaller than the single-particle mean level spacing and the
Bardeen-Cooper-Schrieffer theory of superconductivity breaks down.
Superconductivity and ferromagnetism can coexist in this regime. We identify
signatures of the competition between superconductivity and ferromagnetism in a
number of quantities: ground-state spin, conductance fluctuations when the
grain is weakly coupled to external leads and the thermodynamic properties of
the grain, such as heat capacity and spin susceptibility.Comment: 13 pages, 13 figures, Proceedings of the Conference on the Frontiers
of Quantum and Mesoscopic Thermodynamics (FQMT11
A semismooth newton method for the nearest Euclidean distance matrix problem
The Nearest Euclidean distance matrix problem (NEDM) is a fundamentalcomputational problem in applications such asmultidimensional scaling and molecularconformation from nuclear magnetic resonance data in computational chemistry.Especially in the latter application, the problem is often large scale with the number ofatoms ranging from a few hundreds to a few thousands.In this paper, we introduce asemismooth Newton method that solves the dual problem of (NEDM). We prove that themethod is quadratically convergent.We then present an application of the Newton method to NEDM with -weights.We demonstrate the superior performance of the Newton method over existing methodsincluding the latest quadratic semi-definite programming solver.This research also opens a new avenue towards efficient solution methods for the molecularembedding problem
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