scholarly journals On Accuracy of Improved χ 2-Approximations

2001 ◽  
Vol 8 (2) ◽  
pp. 401-414
Author(s):  
V. V. Ulyanov ◽  
Y. Fujikoshi
Keyword(s):  
Asymptotic Expansion ◽  
Error Bound ◽  
Sufficient Conditions ◽  
Scale Mixture ◽  
Bartlett Correction

Abstract For a statistic S whose distribution can be approximated by χ 2-distributions, there is a considerable interest in constructing improved χ 2-approximations. A typical approach is to consider a transformation T = T(S) based on the Bartlett correction or a Bartlett type correction. In this paper we consider two cases in which S is expressed as a scale mixture of a χ 2-variate or the distribution of S allows an asymptotic expansion in terms of χ 2-distributions. For these statistics, we give sufficient conditions for T to have an improved χ 2-approximation. Furthermore, we present a method for obtaining its error bound.

scholarly journals Exact Penalization in Stochastic Programming—Calmness and Constraint Qualification

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Martin Branda
Keyword(s):  
Stochastic Programming ◽  
Error Bound ◽  
Constraint Qualification ◽  
Sufficient Conditions ◽  
Discrete Distributions ◽  
Exact Penalty ◽  
Asymptotic Equivalence ◽  
Exact Penalization ◽  
Local Minimizers ◽  
Constrained Problems

We deal with the conditions which ensure exact penalization in stochastic programming problems under finite discrete distributions. We give several sufficient conditions for problem calmness including graph calmness, existence of an error bound, and generalized Mangasarian-Fromowitz constraint qualification. We propose a new version of the theorem on asymptotic equivalence of local minimizers of chance constrained problems and problems with exact penalty objective. We apply the theory to a problem with a stochastic vanishing constraint.

Asymptotics of the Titchmarsh-Weyl m-coefficient for integrable potentials

1986 ◽  
Vol 103 (3-4) ◽  
pp. 347-358 ◽  
Author(s):  
Hans G. Kaper ◽  
Man Kam Kwong
Keyword(s):  
Asymptotic Expansion ◽  
Power Series ◽  
Asymptotic Behaviour ◽  
Potential Function ◽  
Error Bound ◽  
Power Series Expansion ◽  
Smoothness Condition ◽  
Asymptotic Power ◽  
Simple Method ◽  
Expansion Coefficients

This article is concerned with the asymptotic behaviour of m(λ), the Titchmarsh-Weyl m-coefficient, for the singular eigenvalue equation y“ + (λ − q(x))y = 0 on [0, ∞), as λ →∞ in a sector in the upper half of the complex plane. It is assumed that the potential function q is integrable near 0. A simplified proof is given of a result of Atkinson [7], who derived the first two terms in the asymptotic expansion of m(λ), and a sharper error bound is obtained. Theproof is then generalised to derive subsequent terms in the asymptotic expansion. It is shown that the Titchmarsh-Weyl m-coefficient admits an asymptotic power series expansion if the potential function satisfies some smoothness condition. A simple method to compute the expansion coefficients is presented. The results for the first few coefficients agree with those given by Harris [9].

ROBUST SYNCHRONIZATION OF CHAOTIC LUR'E SYSTEMS VIA REPLACING VARIABLES CONTROL

2006 ◽  
Vol 16 (11) ◽  
pp. 3421-3433 ◽  
Author(s):  
XIAOFENG WU ◽  
MUHONG WANG
Keyword(s):  
Frequency Domain ◽  
Error Bound ◽  
Sufficient Conditions ◽  
Synchronization Error ◽  
Linear Matrix ◽  
Lur’E Systems ◽  
Parameter Mismatch ◽  
Chaotic Lur’E Systems ◽  
Definition Of ◽  
Synchronization Scheme

The sufficient conditions for chaos synchronization of two nonidentical systems by replacing variables control have not been proposed until now. In this paper, synchronization of two chaotic Lur'e systems with parameter mismatch by replacing variables control is studied. First of all, we present a master-slave Lur'e systems synchronization scheme with both parameter mismatch and replacing variables control, and derive a responsive error system for the scheme. A new definition of synchronization with finite L 2-gain is then introduced. Based on the definition, the sufficient synchronization criteria which are in the form of linear matrix inequality (LMI) are proved using a quadratic Lyapunov function. By means of MKY lemma the frequency domain criteria are further derived from the obtained LMIs. These frequency domain criteria are illustrated on the master-slave Chua's circuits with parameter mismatch so that the ranges of the parameters of Chua's circuit are analytically solved in the sense of the synchronization with finite L 2-gain by replacing singe-variable control. The illustrative examples verify that within the ranges of the parameters it is possible to synchronize the master-slave Chua's circuits up to a small synchronization error bound, even the qualitative behaviors of the slave circuit are different from that of the master one, such as the trajectory of the master circuit is chaotic and that of the slave divergent. The relation between the synchronization error bound and parameter mismatch is shown.

scholarly journals Error Bounds and Finite Termination for Constrained Optimization Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Wenling Zhao ◽  
Daojin Song ◽  
Bingzhuang Liu
Keyword(s):  
Constrained Optimization ◽  
Error Bound ◽  
Feasible Solution ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Merit Function ◽  
Global Error Bound ◽  
Constrained Optimization Problems ◽  
Nonconvex Constrained Optimization ◽  
Necessary And Sufficient

We present a global error bound for the projected gradient of nonconvex constrained optimization problems and a local error bound for the distance from a feasible solution to the optimal solution set of convex constrained optimization problems, by using the merit function involved in the sequential quadratic programming (SQP) method. For the solution sets (stationary points set andKKTpoints set) of nonconvex constrained optimization problems, we establish the definitions of generalized nondegeneration and generalized weak sharp minima. Based on the above, the necessary and sufficient conditions for a feasible solution of the nonconvex constrained optimization problems to terminate finitely at the two solutions are given, respectively. Accordingly, the results in this paper improve and popularize existing results known in the literature. Further, we utilize the global error bound for the projected gradient with the merit function being computed easily to describe these necessary and sufficient conditions.

scholarly journals Strongly regular points of mappings

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Malek Abbasi ◽  
Michel Théra
Keyword(s):  
Error Bound ◽  
Directional Derivative ◽  
Sufficient Conditions ◽  
Strong Regularity ◽  
Strongly Regular

AbstractIn this paper, we use a robust lower directional derivative and provide some sufficient conditions to ensure the strong regularity of a given mapping at a certain point. Then, we discuss the Hoffman estimation and achieve some results for the estimate of the distance to the set of solutions to a system of linear equalities. The advantage of our estimate is that it allows one to calculate the coefficient of the error bound.

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