Perturbation analysis of global error bounds for systems of linear inequalities

This paper studies the existence of a uniform global error bound when a system of linear inequalities is under local arbitrary perturbations. Specifically, given a possibly infinite system of linear inequalities satisfying the Slater’s condition and a certain compactness condition, it is shown that for sufficiently small arbitrary perturbations the perturbed system is solvable and there exists a uniform global error bound if and only if the original system is bounded or its homogeneous system has a strict solution. Received: April 12, 1998 / Accepted: February 11, 2000?Published online July 20, 2000     

Error bounds for analytic systems and their applications

Using a 1958 result of Lojasiewicz, we establish an error bound for analytic systems consisting of equalities and inequalities defined by real analytic functions. In particular, we show that over any bounded region, the distance from any vectorx in the region to the solution set of an analytic system is bounded by a residual function, raised to a certain power, evaluated atx. For quadratic systems satisfying certain nonnegativity assumptions, we show that this exponent is equal to 1/2. We apply the error bounds to the Karush—Kuhn—Tucker system of a variational inequality, the affine variational inequality, the linear and nonlinear complementarity problem, and the 0–1 integer feasibility problem, and obtain new error bound results for these problems. The latter results extend previous work for polynomial systems and explain why a certain square-root term is needed in an error bound for the (monotone) linear complementarity problem.The research of this author is based on work supported by the Natural Sciences and Engineering Research Council of Canada under grant OPG0090391.The research of this author is based on work supported by the National Science Foundation under grants DDM-9104078 and CCR-9213739 and by the Office of Naval Research under grant 4116687-01.     

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.     

Coefficient bounds for some families of starlike and convex functions of complex order

In the present work, the authors determine coefficient bounds for functions in certain subclasses of starlike and convex functions of complex order, which are introduced here by means of a family of nonhomogeneous Cauchy–Euler differential equations. Several corollaries and consequences of the main results are also considered.     

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.     

Merit functions and error bounds for constrained mixed set-valued variational inequalities via generalized f-projection operators

In this paper, we introduce and investigate a constrained mixed set-valued variational inequality (MSVI) in Hilbert spaces. We prove the solution set of the constrained MSVI is a singleton under strict monotonicity. We also propose four merit functions for the constrained MSVI, that is, the natural residual, gap function, regularized gap function and D-gap function. We further use these functions to obtain error bounds, i.e. upper estimates for the distance to solutions of the constrained MSVI under strong monotonicity and Lipschitz continuity. The approach exploited in this paper is based on the generalized f-projection operator due to Wu and Huang, but not the well-known proximal mapping.     

Perturbation analysis and condition numbers of scaled total least squares problems

The standard approaches to solving an overdetermined linear system Ax ≈ b find minimal corrections to the vector b and/or the matrix A such that the corrected system is consistent, such as the least squares (LS), the data least squares (DLS) and the total least squares (TLS). The scaled total least squares (STLS) method unifies the LS, DLS and TLS methods. The classical normwise condition numbers for the LS problem have been widely studied. However, there are no such similar results for the TLS and the STLS problems. In this paper, we first present a perturbation analysis of the STLS problem, which is a generalization of the TLS problem, and give a normwise condition number for the STLS problem. Different from normwise condition numbers, which measure the sizes of both input perturbations and output errors using some norms, componentwise condition numbers take into account the relation of each data component, and possible data sparsity. Then in this paper we give explicit expressions for the estimates of the mixed and componentwise condition numbers for the STLS problem. Since the TLS problem is a special case of the STLS problem, the condition numbers for the TLS problem follow immediately from our STLS results. All the discussions in this paper are under the Golub-Van Loan condition for the existence and uniqueness of the STLS solution. Yimin Wei is supported by the National Natural Science Foundation of China under grant 10871051, Shanghai Science & Technology Committee under grant 08DZ2271900 and Shanghai Education Committee under grant 08SG01. Sanzheng Qiao is partially supported by Shanghai Key Laboratory of Contemporary Applied Mathematics of Fudan University during his visiting.     

A nonlinear inequality and application to global asymptotic stability of perturbed systems

The goal of this work is to present a new nonlinear inequality which is used in a study of the Lyapunov uniform stability and uniform asymptotic stability of solutions to time‐varying perturbed differential equations. New sufficient conditions for global uniform asymptotic stability and/or practical stability in terms of Lyapunov‐like functions for nonlinear time‐varying systems is obtained. Our conditions are expressed as relation between the Lyapunov function and the existence of specific function which appear in our analysis through the solution of a scalar differential equation. Moreover, an example in dimensional two is given to illustrate the applicability of the main result. Copyright © 2014 John Wiley & Sons, Ltd.     

Parametric error bounds for convex approximations of two-stage mixed-integer recourse models with a random second-stage cost vector

We consider two-stage recourse models with integer restrictions in the second stage. These models are typically non-convex and hence, hard to solve. There exist convex approximations of these models with accompanying error bounds. However, it is unclear how these error bounds depend on the distributions of the second-stage cost vector q. In this paper, we derive parametric error bounds whose dependence on the distribution of q is explicit: they scale linearly in the expected value of the ${?}_1$-norm of q.     

Inclusion relations and convolution properties of a certain class of analytic functions associated with the Ruscheweyh derivatives

Let A denote the class of functions f(z) with

f(0)=f^′(0)−1=0,     

Consistency of perturbation analysis for a queue with finite buffer space and loss policy

The subject of discrete-event dynamical systems has taken on a new direction with the advent of perturbation analysis (PA), an efficient method for estimating the gradients of a steady-state performance measure, by analyzing data obtained from a single-simulation experiment in the time domain. A crucial issue is whether PA gives strongly consistent estimates, namely, whether average time-domain-based gradients converge, over infinite horizon, to the steady-state gradients. In this paper, we investigate this issue for a queue with a finite buffer capacity and a loss policy. The performance measure in question is the average amount of lost customers, as a function of the buffer's capacity, which is assumed to be continuous in our work. It is shown that PA gives strongly consistent estimates. The analysis uses a new technique, based on busy period-dependent inequalities. This technique may have possible extensions to analyses of consistency of PA for more general queueing systems.     

Non-uniform robust global asymptotic stability for discrete-time systems and applications to numerical analysis

The notion of non-uniform Robust Global Asymptotic Stability(RGAS) presented in this paper generalizes the notion of non-uniformin time RGAS for finite- or infinite-dimensional discrete-timesystems. Lyapunov characterizations for this stability notionare provided. The results are applied to finite-dimensionaldiscrete-time systems obtained by time discretization of continuous-timesystems by the explicit Euler method.     

Gap Functions and Existence of Solutions for a System of Vector Equilibrium Problems

In this paper, a gap function for a system of vector equilibrium problems is introduced and studied. Some necessary and sufficient conditions for the system of vector equilibrium problems are established. Characterizations of the solutions set for the system of vector equilibrium problems are also derived. Furthermore, some existence results of solutions for the system of vector equilibrium problems are proved. This work was supported by the National Natural Science Foundation of China, the Youth Foundation, Sichuan Education Department of China, the National Natural Science Foundation, Sichuan Education Department of China (2004C018), and a grant from the National Science Council of ROC.     

A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems

We present a generalization of the Kalman rank condition to the case of n × n linear parabolic systems with constant coefficients and diagonalizable diffusion matrix. To reach the result, we are led to prove a global Carleman estimate for the solutions of a scalar 2n-order parabolic equation and deduce from it an observability inequality for our adjoint system. G.-B. Manuel was supported by D.G.E.S. (Spain), grant MTM2006-07932.     

A new methodology for sensitivity and stability analysis of analytic network models

In this paper we develop a methodology to study the sensitivity and the stability of models built using the Analytic Network Process. We study two types of stability: core and solution stability. The former deals with finding the region of the perturbation space in which the initial solution (i.e., the alternative that is ranked first) obtained from the ANP model remains most preferred. The latter deals with finding the regions of the perturbation space in which the solutions that were not initially most preferred (i.e., alternatives that were not ranked first) become most preferred (i.e., they are ranked first). The methodology consists of three stages: generation of the perturbation space, finding the boundaries of the regions in the perturbation space in which the different alternatives are ranked first, and finding the stability regions.     

Stability analysis and control synthesis for a class of switched neutral systems

In this paper, the problems of asymptotical stability and stabilization of a class of switched neutral control systems are investigated. A delay-dependent stability criterion is formulated in term of linear matrix inequalities (LMIs) by using quadratic Lyapunov functions and inequality analysis technique. The corresponding switching rule is obtained through dividing the state space properly. Also, the synthesis of stabilizing state-feedback controllers are done such that the close-loop system is asymptotically stable. Two numerical examples are given to show the proposed method.     

Non-probabilistic convex models and interval analysis method for dynamic response of a beam with bounded uncertainty

This paper is concerned with the comparison of two non-probabilistic set-theoretical models for dynamic response measures of an infinitely long beam. The beam is on an uncertain foundation and subjected to a moving force with constant speed. The steady state vibration is analyzed with finite element method. The dynamic responses of the beam are approximated to the first-order respect of the uncertainty variables. As a rule, in convex models and interval analysis, the uncertainties are considered to be unknown, but they give out their allowable vector space. Comparing the convex models with interval analysis in mathematical proofs and numerical calculations, it’s shows that under the condition of transform an interval vector to an outer enclosed ellipsoid, the dynamic response of the infinitely long beam predicted by interval analysis is smaller than that by convex models; under the condition of transform a hyperellipsoid to an outer enclosed interval vector, the dynamic response of the infinitely long beam calculated by convex models is smaller than that by interval analysis method.     

Perturbation and error analyses for block downdating of a Cholesky decomposition

A new perturbation result is presented for the problem of block downdating a Cholesky decompositionX ^T X = R ^T R. Then, a condition number for block downdating is proposed and compared to other downdating condition numbers presented in literature recently. This new condition number is shown to give a tighter bound in many cases. Using the perturbation theory, an error analysis is presented for the block downdating algorithms based on the LINPACK downdating algorithm and stabilized hyperbolic transformations. An error analysis is also given for block downdating using Corrected Seminormal Equations (CSNE), and it is shown that for ill-conditioned downdates this method gives more accurate results than the algorithms based on the LINPACK downdating algorithm or hyperbolic transformations. We classify the problems for which the CSNE downdating method produces a downdated upper triangular matrix which is comparable in accuracy to the upper triangular factor obtained from the QR decomposition by Householder transformations on the data matrix with the row block deleted.Dedicated to Ji-guang Sun in honour of his 60th birthdayThe work of the second author was supported in part by the National Science Foundation grant CCR-9209726.     

Existence and global bounds for a fluid model of plasma display technology

This article considers the fluid model for the discharge of plasma particle species in display technology. The fluid equations are coupled with Poisson's equation, which describes the effect of the charged particles on the electric field. The diffusion and mobility coefficients for the positive ion particles depend on the electric field, while those for the electrons depend on the electron mean energy. The reaction rates are proportional to the products of the densities of the reacting particles involved in the particular ionization, conversion or recombination reactions. Moreover, the ionization coefficients are dependent on the electric field, which varies spatially and temporally. The main ionization and discharge reactions are described by an initial-boundary value problem for a system of coupled parabolic–elliptic partial differential equations. The system is first analyzed by upper–lower solution method. By means of the a priori bounds obtained for an arbitrary time, the existence of solution for the initial-boundary value problem is proved in an appropriate Hölder space.     

Inclusion and argument properties for certain subclasses of meromorphic functions associated with a family of multiplier transformations

Making use of a multiplier transformation, which is defined here by means of the Hadamard product (or convolution), the authors introduce some new subclasses of meromorphic functions and investigate their inclusion relationships and argument properties. Some integral-preserving properties in a given sector are also considered.