We propose a branch-and-bound framework for the global optimization of unconstrained holder functions. The general framework is used to derive two algorithms. The first one is a generalization of Piyavskii's algor...
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We propose a branch-and-bound framework for the global optimization of unconstrained holder functions. The general framework is used to derive two algorithms. The first one is a generalization of Piyavskii's algorithm for univariate Lipschitz functions. The second algorithm, using a piecewise constant upper-bounding function, is designed for multivariate holder functions. A proof of convergence is provided for both algorithms. Computational experience is reported on several test functions from the literature.
We extend the Stieltjes integral to holder functions of two variables and prove an existence and uniqueness result for the corresponding deterministic ordinary differential equations and also for stochastic equations ...
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We extend the Stieltjes integral to holder functions of two variables and prove an existence and uniqueness result for the corresponding deterministic ordinary differential equations and also for stochastic equations driven by a two-parameter fractional Brownian motion. (C) 2003 Elsevier Inc. All rights reserved.
Let F(alpha) = {f:R --> R: sup(x)\f(x)\ less-than-or-equal-to 1, sup(x not-equal y) \f(x) - f(y)\/\x - y\alpha less-than-or-equal-to 1}. It is shown that the condition SIGMA(k=1)infinity (Pr{k - 1 < Absolute val...
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Let F(alpha) = {f:R --> R: sup(x)\f(x)\ less-than-or-equal-to 1, sup(x not-equal y) \f(x) - f(y)\/\x - y\alpha less-than-or-equal-to 1}. It is shown that the condition SIGMA(k=1)infinity (Pr{k - 1 < Absolute value of X less-than-or-equal-to k})1/2 < infinity, which is known to be equivalent to F1 being P-Donsker, implies that F(alpha) is P-Donsker for 1/2 < alpha < 1. U-processes indexed by these classes of holder's functions are also considered.
This paper deals with the one-dimensional global optimization problem where the objective function satisfies a holder condition over a closed interval. A direct extension of the popular Piyavskii method proposed for L...
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This paper deals with the one-dimensional global optimization problem where the objective function satisfies a holder condition over a closed interval. A direct extension of the popular Piyavskii method proposed for Lipschitz functions to holder optimization requires an a priori estimate of the holder constant and solution to an equation of degree N at each iteration. In this paper a new scheme is introduced. Three algorithms are proposed for solving one-dimensional holder global optimization problems. All of them work without solving equations of degree N. The ease (very often arising in applications) when a holder constant is not given a priori is considered. It is shown that local information about the objective function used inside the global procedure can accelerate the search significantly. Numerical experiments show quite promising performance of the new algorithms.
This paper deals with three classes of functions of great importance in analysis and its applications. We construct a family of holder functions in the closed unit interval having two continuous parameters. Those func...
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This paper deals with three classes of functions of great importance in analysis and its applications. We construct a family of holder functions in the closed unit interval having two continuous parameters. Those functions are not of bounded variation for any pair of values of the holder constant and exponent. The construction depends on a change of variables given by a Lipschitz function with constant equal to 1. Several questions related to the concepts of genericity, surjectivity and deformability are posed at, the end.
In this paper we analyze the oscillation of functions having derivatives in the holder or Zygmund class in terms of generalized differences and prove that its growth is governed by a version of the classical Kolmogoro...
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In this paper we analyze the oscillation of functions having derivatives in the holder or Zygmund class in terms of generalized differences and prove that its growth is governed by a version of the classical Kolmogorov's Law of the Iterated Logarithm. A better behavior is obtained for functions in the Lipschitz class via an interesting connection with Caldern-Zygmund operators.
In this paper, the global optimization problem min(y is an element of S) F(y) with S being a hyperinterval in R-N and F(y) satisfying the Lipschitz condition with an unknown Lipschitz constant is considered. It is sup...
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In this paper, the global optimization problem min(y is an element of S) F(y) with S being a hyperinterval in R-N and F(y) satisfying the Lipschitz condition with an unknown Lipschitz constant is considered. It is supposed that the function F(y) 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 holder condition. Second, the algorithm at each iteration applies a new geometric technique working with a number of possible holder 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 holder 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. (C) 2014 Elsevier B.V. All rights reserved.
Some hydrodynamic phenomena of an underwater Remotely Operated Vehicle (ROV), such as turbulence, cavitation, and multi-phase fluidic regimes, are associated to continuous but nowhere differentiable functions. These d...
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Some hydrodynamic phenomena of an underwater Remotely Operated Vehicle (ROV), such as turbulence, cavitation, and multi-phase fluidic regimes, are associated to continuous but nowhere differentiable functions. These disturbances stand as complex forces potentially influencing the ROVs during typical navigation tasks. In this paper;the tracking control of a ROV subject to rionsmooth holder disturbances is proposed based on a fractional order robust controller that ensures exponential tracking. Notably, the controller gives rise to a closed-loop system with the following characteristics: a) continuous control signal that alleviates chattering effects;b) the fractional sliding motion is substantiated on a proposed resetting memory principle;c) the control is robust to model uncertainties;and d) exact rejection of holder disturbances in finite-time. A representative simulation study reveals the feasibility of the proposed scheme.
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