Computable Error Bounds for Semidefinite Programming

We study computability and applicability of error bounds for a given semidefinite pro-gramming problem under the assumption that the recession function associated with the constraint system satisfies the Slater condition. Specifically, we give computable error bounds for the distances between feasible sets, optimal objective values, and optimal solution sets in terms of an upper bound for the condition number of a constraint system, a Lipschitz constant of the objective function, and the size of perturbation. Moreover, we are able to obtain an exact penalty function for semidefinite programming along with a lower bound for penalty parameters. We also apply the results to a class of statistical problems.     

一类关于正规三角函数基的标准模糊系统逼近误差公式的讨论

在标准模糊系统的基础上提出了以正规三角函数为基函数的一类模糊系统.通过采用数值分析中的余项与辅助函数方法,对该类模糊系统进行了逼近误差精度的分析,给出了从SISO到MISO的误差界公式.最后,指出了这些公式在模糊系统的理论研究与实际应用的意义.     

A class of optimal iterative methods of inverting a linear bounded operator

A family of higher-order iterative methods for inverting a linear bounded operator in a Banach space is presented. This family depending on two integer parameters can be considered as hyperpower methods of [1]. A convergence Theorem is proved and an optimal class with respect to the efficiency index is determined . Several error bounds are derived and compared with those in the literature.     

A Certain Class of Weighted Approximations for Integrable Functions and Applications

A certain class of weighted approximations, which extends the results of Masjed-Jamei [6] is introduced for integrable functions and some of upper bounds are obtained for the absolute value of the errors of such approximations in two L¹[a, b] and L^∞[a, b] spaces. As the main motivation for introducing the aforesaid class, it is shown that many new inequalities can be generated from the given error bounds. Some illustrative examples are presented in this sense. Moreover, by using the obtained error bounds, a nonstandard type of three-point weighted quadrature rules is introduced and its error bounds are computed.     

Convergence of the control parameterization Ritz method for nonlinear optimal control problems

This paper studies the local convergence properties of the control parameterization Ritz method in which the control variable is approximated over a finite-dimensional subspace. The nonlinear free-endpoint optimal control problem is considered, and error bounds are derived for both the cost functional and state-control convergence. Explicit error bounds are obtained for the particular case of approximations over spline spaces. On specializing the general results to the linear-quadratic regulator problem, global convergence results are obtained. Computational results supporting the theoretically derived error bounds are presented.This research was supported by the University Grants Committee of New Zealand.     

Generalized Christoffel functions and error of positive quadrature

The author gives some upper and lower bounds for the generalized Christoffel functions related to a Ditzian-Totik generalized weight. As an application, an error estimate of Gauss quadrature formula inL ₁-weighted norm is derived.Dedicated to Prof. Luigi Gatteschi on the occasion of his 70th birthdayWork sponsored by MURST 40%.     

Gap functions and global error bounds for set-valued variational inequalities

The set-valued variational inequality problem is very useful in economics theory and nonsmooth optimization. In this paper, we introduce some gap functions for set-valued variational inequality problems under suitable assumptions. By using these gap functions we derive global error bounds for the solution of the set-valued variational inequality problems. Our results not only generalize the previously known results for classical variational inequalities from single-valued case to set-valued, but also present a way to construct gap functions and derive global error bounds for set-valued variational inequality problems.     

Computation of Error Bounds for P-matrix Linear Complementarity Problems

We give new error bounds for the linear complementarity problem where the involved matrix is a P-matrix. Computation of rigorous error bounds can be turned into a P-matrix linear interval system. Moreover, for the involved matrix being an H-matrix with positive diagonals, an error bound can be found by solving a linear system of equations, which is sharper than the Mathias-Pang error bound. Preliminary numerical results show that the proposed error bound is efficient for verifying accuracy of approximate solutions. This work is partly supported by a Grant-in-Aid from Japan Society for the Promotion of Science.     

关于两类具有不同基函数的标准模糊系统逼近精度问题的研究

在标准模糊系统的基础上提出了以正规二次多项式和正规三角函数为基函数的两类标准模糊系统.通过采用数值分析中的余项与辅助函数方法,对这两类模糊系统进行了误差精度的分析,给出了从SISO到MISO的误差界公式.同时,对这两类模糊系统误差界进行了比较,指出了两类模糊系统的优劣.最后,通过算例验证了理论结果的正确性.     

Gap functions and error bounds for weak vector variational inequalities

In this article, by using the image space analysis, a gap function for weak vector variational inequalities is obtained. Its lower semicontinuity is also discussed. Then, these results are applied to obtain the error bounds for weak vector variational inequalities. These bounds provide effective estimated distances between a feasible point and the solution set of the weak vector variational inequalities.     

New Uniform Parametric Error Bounds

A uniform parametric error bound is a uniform error estimate for feasible solutions of a family of parametric mathematical programming problems. It has been proven useful in exact penalty formulation for bilevel programming problems. In this paper, we derive new sufficient conditions for the existence of uniform parametric error bounds.     

Quadrature Formulas for Refinable Functions and Wavelets II: Error Analysis

This paper is concerned with the construction and the analysis of Gauss quadrature formulas for computing integrals of (smooth) functions against refinable functions and wavelets. The main goal of this paper is to develop rigorous error estimates for these formulas. For the univariate setting, we derive asymptotic error bounds for a huge class of weight functions including spline functions. We also discuss multivariate quadrature rules and present error estimates for specific nonseparable refinable functions, i.e., for some special box splines.     

The Fourier-series method for inverting transforms of probability distributions

This paper reviews the Fourier-series method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourier-series method are remarkably easy to use, requiring programs of less than fifty lines. The Fourier-series method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule. The same formula is obtained by using the Fourier series of an associated periodic function constructed by aliasing; this explains the name of the method. This Fourier analysis applies to the inversion problem because the Fourier coefficients are just values of the transform. The mathematical centerpiece of the Fourier-series method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it. The greatest difficulty is approximately calculating the infinite series obtained from the inversion integral. Within this framework, lattice cdf's can be calculated from generating functions by finite sums without truncation. For other cdf's, an appropriate truncation of the infinite series can be determined from the transform based on estimates or bounds. For Laplace transforms, the numerical integration can be made to produce a nearly alternating series, so that the convergence can be accelerated by techniques such as Euler summation. Alternatively, the cdf can be perturbed slightly by convolution smoothing or windowing to produce a truncation error bound independent of the original cdf. Although error bounds can be determined, an effective approach is to use two different methods without elaborate error analysis. For this purpose, we also describe two methods for inverting Laplace transforms based on the Post-Widder inversion formula. The overall procedure is illustrated by several queueing examples.     

Gap functions and error bounds for nonsmooth convex vector optimization problem

Abstract

In this article, our main aim is to develop gap functions and error bounds for a (non-smooth) convex vector optimization problem. We show that by focusing on convexity we are able to quite efficiently compute the gap functions and try to gain insight about the structure of set of weak Pareto minimizers by viewing its graph. We will discuss several properties of gap functions and develop error bounds when the data are strongly convex. We also compare our results with some recent results on weak vector variational inequalities with set-valued maps, and also argue as to why we focus on the convex case.     

Refined Error Estimates for Radial Basis Function Interpolation

We discuss new and refined error estimates for radial-function scattered-data interpolants and their derivatives. These estimates hold on R ^d , the d-torus, and the 2-sphere. We employ a new technique, involving norming sets, that enables us to obtain error estimates, which in many cases give bounds orders of magnitude smaller than those previously known.     

Gaussian integration of Chebyshev polynomials and analytic functions

Explicit bounds for the quadrature error of thenth Gauss-Legendre quadrature rule applied to themth Chebyshev polynomial are derived. They are precise up to the orderO(m ⁴ n ^–6). As an application, error constants for classes of functions which are analytic in the interior of an ellipse are estimated. The location of the maxima of the corresponding kernel function is investigated.Dedicated to Luigi Gatteschi on the occasion of his 70th birthday     

Lower Bounds for the Rate of Convergence of Dynamic Reconstruction Algorithms for Distributed-Parameter Systems

We obtain lower bounds for the rate of convergence of reconstruction algorithms for distributed-parameter systems of parabolic type. In the case of a pointwise constraint on control for known reconstruction algorithms, we establish a lower bound on the rate of convergence, which shows that, given certain conditions, for each solution of the system one can choose such a collection of measurements so that the reconstruction error will not be less than a certain value. In the case of unbounded controls, we obtain lower bounds for a possible reconstruction error for each trajectory as well as for a given set of trajectories. For a system of special form, we construct an algorithm for which we obtain upper and lower bounds for accuracy having identical order for a specific choice of matching of the parameters.     

One Approach to the Comparison of Error Bounds at a Point and on a Set in the Solution of Ill-Posed Problems

The approximate solution of ill-posed problems by the regularization method always involves the issue of estimating the error. It is a common practice to use uniform bounds on the whole class of well-posedness in terms of the modulus of continuity of the inverse operator on this class. Local error bounds, which are also called error bounds at a point, have been studied much less. Since the solution of a real-life ill-posed problem is unique, an error bound obtained on the whole class of well-posedness roughens to a great extent the true error bound. In the present paper, we study the difference between error bounds on the class of well-posedness and error bounds at a point for a special class of ill-posed problems. Assuming that the exact solution is a piecewise smooth function, we prove that an error bound at a point is infinitely smaller than the exact bound on the class of well-posedness.     

On the extremality of maximal dual feasible functions

Dual feasible functions have been used to compute bounds and valid inequalities for combinatorial optimization problems. Here, we analyze the properties of some of the best functions proposed so far. Additionally, we provide new results for composed functions. These results will allow improving the computation of bounds and valid inequalities.     

Trigonometric spline and spectral bounds for the solution of linear time-periodic systems

Linear time-periodic systems arise whenever a nonlinear system is linearized about a periodic trajectory. Examples include anisotropic rotor-bearing systems and parametrically excited systems. The structure of the solution to linear time-periodic systems is known due to Floquet’s Theorem. We use this information to derive a new norm which yields two-sided bounds on the solution and in this norm vibrations of the solution are suppressed. The obtained results are a generalization for linear time-invariant systems. Since Floquet’s Theorem is non-constructive, the applicability of the aforementioned results suffers in general from an unknown Floquet normal form. Hence, we discuss trigonometric splines and spectral methods that are both equipped with rigorous bounds on the solution. The methodology differs systematically for the two methods. While in the first method the solution is approximated by trigonometric splines and the upper bound depends on the approximation quality, in the second method the linear time-periodic system is approximated and its solution is represented as an infinite series. Depending on the smoothness of the time-periodic system, we formulate two upper bounds which incorporate the approximation error of the linear time-periodic system and the truncation error of the series representation. Rigorous bounds on the solution are necessary whenever reliable results are needed, and hence they can support the analysis and, e.g., stability or robustness of the solution may be proven or falsified. The theoretical results are illustrated and compared to trigonometric spline bounds and spectral bounds by means of three examples that include an anisotropic rotor-bearing system and a parametrically excited Cantilever beam.