229 research outputs found
Lower bounds on the size of semidefinite programming relaxations
We introduce a method for proving lower bounds on the efficacy of
semidefinite programming (SDP) relaxations for combinatorial problems. In
particular, we show that the cut, TSP, and stable set polytopes on -vertex
graphs are not the linear image of the feasible region of any SDP (i.e., any
spectrahedron) of dimension less than , for some constant .
This result yields the first super-polynomial lower bounds on the semidefinite
extension complexity of any explicit family of polytopes.
Our results follow from a general technique for proving lower bounds on the
positive semidefinite rank of a matrix. To this end, we establish a close
connection between arbitrary SDPs and those arising from the sum-of-squares SDP
hierarchy. For approximating maximum constraint satisfaction problems, we prove
that SDPs of polynomial-size are equivalent in power to those arising from
degree- sum-of-squares relaxations. This result implies, for instance,
that no family of polynomial-size SDP relaxations can achieve better than a
7/8-approximation for MAX-3-SAT
Field theoretic approach to the counting problem of Hamiltonian cycles of graphs
A Hamiltonian cycle of a graph is a closed path that visits each site once
and only once. I study a field theoretic representation for the number of
Hamiltonian cycles for arbitrary graphs. By integrating out quadratic
fluctuations around the saddle point, one obtains an estimate for the number
which reflects characteristics of graphs well. The accuracy of the estimate is
verified by applying it to 2d square lattices with various boundary conditions.
This is the first example of extracting meaningful information from the
quadratic approximation to the field theory representation.Comment: 5 pages, 3 figures, uses epsf.sty. Estimates for the site entropy and
the gamma exponent indicated explicitl
The Effect of Neutral Atoms on Capillary Discharge Z-pinch
We study the effect of neutral atoms on the dynamics of a capillary discharge
Z-pinch, in a regime for which a large soft-x-ray amplification has been
demonstrated. We extended the commonly used one-fluid magneto-hydrodynamics
(MHD) model by separating out the neutral atoms as a second fluid. Numerical
calculations using this extended model yield new predictions for the dynamics
of the pinch collapse, and better agreement with known measured data.Comment: 4 pages, 4 postscript figures, to be published in Phys. Rev. Let
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
A new picture of the Lifshitz critical behavior
New field theoretic renormalization group methods are developed to describe
in a unified fashion the critical exponents of an m-fold Lifshitz point at the
two-loop order in the anisotropic (m not equal to d) and isotropic (m=d close
to 8) situations. The general theory is illustrated for the N-vector phi^4
model describing a d-dimensional system. A new regularization and
renormalization procedure is presented for both types of Lifshitz behavior. The
anisotropic cases are formulated with two independent renormalization group
transformations. The description of the isotropic behavior requires only one
type of renormalization group transformation. We point out the conceptual
advantages implicit in this picture and show how this framework is related to
other previous renormalization group treatments for the Lifshitz problem. The
Feynman diagrams of arbitrary loop-order can be performed analytically provided
these integrals are considered to be homogeneous functions of the external
momenta scales. The anisotropic universality class (N,d,m) reduces easily to
the Ising-like (N,d) when m=0. We show that the isotropic universality class
(N,m) when m is close to 8 cannot be obtained from the anisotropic one in the
limit d --> m near 8. The exponents for the uniaxial case d=3, N=m=1 are in
good agreement with recent Monte Carlo simulations for the ANNNI model.Comment: 48 pages, no figures, two typos fixe
Implementation of an Optimal First-Order Method for Strongly Convex Total Variation Regularization
We present a practical implementation of an optimal first-order method, due
to Nesterov, for large-scale total variation regularization in tomographic
reconstruction, image deblurring, etc. The algorithm applies to -strongly
convex objective functions with -Lipschitz continuous gradient. In the
framework of Nesterov both and are assumed known -- an assumption
that is seldom satisfied in practice. We propose to incorporate mechanisms to
estimate locally sufficient and during the iterations. The mechanisms
also allow for the application to non-strongly convex functions. We discuss the
iteration complexity of several first-order methods, including the proposed
algorithm, and we use a 3D tomography problem to compare the performance of
these methods. The results show that for ill-conditioned problems solved to
high accuracy, the proposed method significantly outperforms state-of-the-art
first-order methods, as also suggested by theoretical results.Comment: 23 pages, 4 figure
Scaling of Self-Avoiding Walks in High Dimensions
We examine self-avoiding walks in dimensions 4 to 8 using high-precision
Monte-Carlo simulations up to length N=16384, providing the first such results
in dimensions on which we concentrate our analysis. We analyse the
scaling behaviour of the partition function and the statistics of
nearest-neighbour contacts, as well as the average geometric size of the walks,
and compare our results to -expansions and to excellent rigorous bounds
that exist. In particular, we obtain precise values for the connective
constants, , , ,
and give a revised estimate of . All of
these are by at least one order of magnitude more accurate than those
previously given (from other approaches in and all approaches in ).
Our results are consistent with most theoretical predictions, though in
we find clear evidence of anomalous -corrections for the scaling of
the geometric size of the walks, which we understand as a non-analytic
correction to scaling of the general form (not present in pure
Gaussian random walks).Comment: 14 pages, 2 figure
New Lower Bounds on the Self-Avoiding-Walk Connective Constant
We give an elementary new method for obtaining rigorous lower bounds on the
connective constant for self-avoiding walks on the hypercubic lattice .
The method is based on loop erasure and restoration, and does not require exact
enumeration data. Our bounds are best for high , and in fact agree with the
first four terms of the expansion for the connective constant. The bounds
are the best to date for dimensions , but do not produce good results
in two dimensions. For , respectively, our lower bound is within
2.4\%, 0.43\%, 0.12\%, 0.044\% of the value estimated by series extrapolation.Comment: 35 pages, 388480 bytes Postscript, NYU-TH-93/02/0
Entropy of chains placed on the square lattice
We obtain the entropy of flexible linear chains composed of M monomers placed
on the square lattice using a transfer matrix approach. An excluded volume
interaction is included by considering the chains to be self-and mutually
avoiding, and a fraction rho of the sites are occupied by monomers. We solve
the problem exactly on stripes of increasing width m and then extrapolate our
results to the two-dimensional limit to infinity using finite-size scaling. The
extrapolated results for several finite values of M and in the polymer limit M
to infinity for the cases where all lattice sites are occupied (rho=1) and for
the partially filled case rho<1 are compared with earlier results. These
results are exact for dimers (M=2) and full occupation (\rho=1) and derived
from series expansions, mean-field like approximations, and transfer matrix
calculations for some other cases. For small values of M, as well as for the
polymer limit M to infinity, rather precise estimates of the entropy are
obtained.Comment: 6 pages, 7 figure
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