677 research outputs found
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 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
Application of reduced-set pareto-lipschitzian optimization to truss optimization
In this paper, a recently proposed global Lipschitz optimization algorithm Pareto-Lipschitzian Optimization with Reduced-set (PLOR) is further developed, investigated and applied to truss optimization problems. Partition patterns of the PLOR algorithm are similar to those of DIviding RECTangles (DIRECT), which was widely applied to different real-life problems. However here a set of all Lipschitz constants is reduced to just two: the maximal and the minimal ones. In such a way the PLOR approach is independent of any user-defined parameters and balances equally local and global search during the optimization process. An expanded list of other well-known DIRECT-type algorithms is used in investigation and experimental comparison using the standard test problems and truss optimization problems. The experimental investigation shows that the PLOR algorithm gives very competitive results to other DIRECT-type algorithms using standard test problems and performs pretty well on real truss optimization problems
Lipschitz Optimisation for Lipschitz Interpolation
Techniques known as Nonlinear Set Membership prediction, Kinky Inference or
Lipschitz Interpolation are fast and numerically robust approaches to
nonparametric machine learning that have been proposed to be utilised in the
context of system identification and learning-based control. They utilise
presupposed Lipschitz properties in order to compute inferences over unobserved
function values. Unfortunately, most of these approaches rely on exact
knowledge about the input space metric as well as about the Lipschitz constant.
Furthermore, existing techniques to estimate the Lipschitz constants from the
data are not robust to noise or seem to be ad-hoc and typically are decoupled
from the ultimate learning and prediction task. To overcome these limitations,
we propose an approach for optimising parameters of the presupposed metrics by
minimising validation set prediction errors. To avoid poor performance due to
local minima, we propose to utilise Lipschitz properties of the optimisation
objective to ensure global optimisation success. The resulting approach is a
new flexible method for nonparametric black-box learning. We provide
experimental evidence of the competitiveness of our approach on artificial as
well as on real data
New technique for solving univariate global optimization
summary:In this paper, a new global optimization method is proposed for an optimization problem with twice differentiable objective function a single variable with box constraint. The method employs a difference of linear interpolant of the objective and a concave function, where the former is a continuous piecewise convex quadratic function underestimator. The main objectives of this research are to determine the value of the lower bound that does not need an iterative local optimizer. The proposed method is proven to have a finite convergence to locate the global optimum point. The numerical experiments indicate that the proposed method competes with another covering methods
Let's Make Block Coordinate Descent Go Fast: Faster Greedy Rules, Message-Passing, Active-Set Complexity, and Superlinear Convergence
Block coordinate descent (BCD) methods are widely-used for large-scale
numerical optimization because of their cheap iteration costs, low memory
requirements, amenability to parallelization, and ability to exploit problem
structure. Three main algorithmic choices influence the performance of BCD
methods: the block partitioning strategy, the block selection rule, and the
block update rule. In this paper we explore all three of these building blocks
and propose variations for each that can lead to significantly faster BCD
methods. We (i) propose new greedy block-selection strategies that guarantee
more progress per iteration than the Gauss-Southwell rule; (ii) explore
practical issues like how to implement the new rules when using "variable"
blocks; (iii) explore the use of message-passing to compute matrix or Newton
updates efficiently on huge blocks for problems with a sparse dependency
between variables; and (iv) consider optimal active manifold identification,
which leads to bounds on the "active set complexity" of BCD methods and leads
to superlinear convergence for certain problems with sparse solutions (and in
some cases finite termination at an optimal solution). We support all of our
findings with numerical results for the classic machine learning problems of
least squares, logistic regression, multi-class logistic regression, label
propagation, and L1-regularization
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