1,641 research outputs found
Solving satisfiability problems by fluctuations: The dynamics of stochastic local search algorithms
Stochastic local search algorithms are frequently used to numerically solve
hard combinatorial optimization or decision problems. We give numerical and
approximate analytical descriptions of the dynamics of such algorithms applied
to random satisfiability problems. We find two different dynamical regimes,
depending on the number of constraints per variable: For low constraintness,
the problems are solved efficiently, i.e. in linear time. For higher
constraintness, the solution times become exponential. We observe that the
dynamical behavior is characterized by a fast equilibration and fluctuations
around this equilibrium. If the algorithm runs long enough, an exponentially
rare fluctuation towards a solution appears.Comment: 21 pages, 18 figures, revised version, to app. in PRE (2003
A distance on curves modulo rigid transformations
We propose a geometric method for quantifying the difference between
parametrized curves in Euclidean space by introducing a distance function on
the space of parametrized curves up to rigid transformations (rotations and
translations). Given two curves, the distance between them is defined as the
infimum of an energy functional which, roughly speaking, measures the extent to
which the jet field of the first curve needs to be rotated to match up with the
jet field of the second curve. We show that this energy functional attains a
global minimum on the appropriate function space, and we derive a set of
first-order ODEs for the minimizer.Comment: 22 pages, 1 figure; final version as published with minor typos
correcte
Gather-and-broadcast frequency control in power systems
We propose a novel frequency control approach in between centralized and
distributed architectures, that is a continuous-time feedback control version
of the dual decomposition optimization method. Specifically, a convex
combination of the frequency measurements is centrally aggregated, followed by
an integral control and a broadcast signal, which is then optimally allocated
at local generation units. We show that our gather-and-broadcast control
architecture comprises many previously proposed strategies as special cases. We
prove local asymptotic stability of the closed-loop equilibria of the
considered power system model, which is a nonlinear differential-algebraic
system that includes traditional generators, frequency-responsive devices, as
well as passive loads, where the sources are already equipped with primary
droop control. Our feedback control is designed such that the closed-loop
equilibria of the power system solve the optimal economic dispatch problem
Shell-crossing in quasi-one-dimensional flow
Blow-up of solutions for the cosmological fluid equations, often dubbed
shell-crossing or orbit crossing, denotes the breakdown of the single-stream
regime of the cold-dark-matter fluid. At this instant, the velocity becomes
multi-valued and the density singular. Shell-crossing is well understood in one
dimension (1D), but not in higher dimensions. This paper is about
quasi-one-dimensional (Q1D) flow that depends on all three coordinates but
differs only slightly from a strictly 1D flow, thereby allowing a perturbative
treatment of shell-crossing using the Euler--Poisson equations written in
Lagrangian coordinates. The signature of shell-crossing is then just the
vanishing of the Jacobian of the Lagrangian map, a regular perturbation
problem. In essence the problem of the first shell-crossing, which is highly
singular in Eulerian coordinates, has been desingularized by switching to
Lagrangian coordinates, and can then be handled by perturbation theory. Here,
all-order recursion relations are obtained for the time-Taylor coefficients of
the displacement field, and it is shown that the Taylor series has an infinite
radius of convergence. This allows the determination of the time and location
of the first shell-crossing, which is generically shown to be taking place
earlier than for the unperturbed 1D flow. The time variable used for these
statements is not the cosmic time but the linear growth time . For simplicity, calculations are restricted to an Einstein--de Sitter
universe in the Newtonian approximation, and tailored initial data are used.
However it is straightforward to relax these limitations, if needed.Comment: 9 pages; received 2017 May 24, and accepted 2017 June 21 at MNRA
Relaxation and Metastability in the RandomWalkSAT search procedure
An analysis of the average properties of a local search resolution procedure
for the satisfaction of random Boolean constraints is presented. Depending on
the ratio alpha of constraints per variable, resolution takes a time T_res
growing linearly (T_res \sim tau(alpha) N, alpha < alpha_d) or exponentially
(T_res \sim exp(N zeta(alpha)), alpha > alpha_d) with the size N of the
instance. The relaxation time tau(alpha) in the linear phase is calculated
through a systematic expansion scheme based on a quantum formulation of the
evolution operator. For alpha > alpha_d, the system is trapped in some
metastable state, and resolution occurs from escape from this state through
crossing of a large barrier. An annealed calculation of the height zeta(alpha)
of this barrier is proposed. The polynomial/exponentiel cross-over alpha_d is
not related to the onset of clustering among solutions.Comment: 23 pages, 11 figures. A mistake in sec. IV.B has been correcte
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