8,582 research outputs found
Lipschitz gradients for global optimization in a one-point-based partitioning scheme
A global optimization problem is studied where the objective function
is a multidimensional black-box function and its gradient satisfies the
Lipschitz condition over a hyperinterval with an unknown Lipschitz constant
. Different methods for solving this problem by using an a priori given
estimate of , its adaptive estimates, and adaptive estimates of local
Lipschitz constants are known in the literature. Recently, the authors have
proposed a one-dimensional algorithm working with multiple estimates of the
Lipschitz constant for (the existence of such an algorithm was a
challenge for 15 years). In this paper, a new multidimensional geometric method
evolving the ideas of this one-dimensional scheme and using an efficient
one-point-based partitioning strategy is proposed. Numerical experiments
executed on 800 multidimensional test functions demonstrate quite a promising
performance in comparison with popular DIRECT-based methods.Comment: 25 pages, 4 figures, 5 tables. arXiv admin note: text overlap with
arXiv:1103.205
Deterministic global optimization using space-filling curves and multiple estimates of Lipschitz and Holder constants
In this paper, the global optimization problem with
being a hyperinterval in and satisfying the Lipschitz condition
with an unknown Lipschitz constant is considered. It is supposed that the
function can be multiextremal, non-differentiable, and given as a
`black-box'. To attack the problem, a new global optimization algorithm based
on the following two ideas is proposed and studied both theoretically and
numerically. First, the new algorithm uses numerical approximations to
space-filling curves to reduce the original Lipschitz multi-dimensional problem
to a univariate one satisfying the H\"{o}lder condition. Second, the algorithm
at each iteration applies a new geometric technique working with a number of
possible H\"{o}lder constants chosen from a set of values varying from zero to
infinity showing so that ideas introduced in a popular DIRECT method can be
used in the H\"{o}lder global optimization. Convergence conditions of the
resulting deterministic global optimization method are established. Numerical
experiments carried out on several hundreds of test functions show quite a
promising performance of the new algorithm in comparison with its direct
competitors.Comment: 26 pages, 10 figures, 4 table
Lipschitz optimization methods for fitting a sum of damped sinusoids to a series of observations
A general nonlinear regression model is considered in the form of fitting a sum of damped sinusoids to a series of non-uniform observations. The problem of parameter estimation in this model is important in many applications like signal processing. The corresponding continuous optimization problem is typically difficult due to the high multiextremal character of the objective function. It is shown how Lipschitz-based deterministic methods can be well-suited for studying these challenging global optimization problems, when a limited computational budget is given and some guarantee of the found solution is required
Selection and Estimation for Mixed Graphical Models
We consider the problem of estimating the parameters in a pairwise graphical
model in which the distribution of each node, conditioned on the others, may
have a different parametric form. In particular, we assume that each node's
conditional distribution is in the exponential family. We identify restrictions
on the parameter space required for the existence of a well-defined joint
density, and establish the consistency of the neighbourhood selection approach
for graph reconstruction in high dimensions when the true underlying graph is
sparse. Motivated by our theoretical results, we investigate the selection of
edges between nodes whose conditional distributions take different parametric
forms, and show that efficiency can be gained if edge estimates obtained from
the regressions of particular nodes are used to reconstruct the graph. These
results are illustrated with examples of Gaussian, Bernoulli, Poisson and
exponential distributions. Our theoretical findings are corroborated by
evidence from simulation studies
Quantum Thermodynamics
Quantum thermodynamics addresses the emergence of thermodynamical laws from
quantum mechanics. The link is based on the intimate connection of quantum
thermodynamics with the theory of open quantum systems. Quantum mechanics
inserts dynamics into thermodynamics giving a sound foundation to
finite-time-thermodynamics. The emergence of the 0-law I-law II-law and III-law
of thermodynamics from quantum considerations is presented. The emphasis is on
consistence between the two theories which address the same subject from
different foundations. We claim that inconsistency is the result of faulty
analysis pointing to flaws in approximations
- …