Convergence of an Interior Point Algorithm for Continuous Minimax

We propose an algorithm for the constrained continuous minimax problem. The algorithm uses a quasi-Newton search direction, based on subgradient information, conditional on maximizers. The initial problem is transformed to an equivalent equality constrained problem, where the logarithmic barrier function is used to ensure feasibility. In the case of multiple maximizers, the algorithm adopts semi-infinite programming iterations toward epiconvergence. Satisfaction of the equality constraints is ensured by an adaptive quadratic penalty function. The algorithm is augmented by a discrete minimax procedure to compute the semi-infinite programming steps and ensure overall progress when required by the adaptive penalty procedure. Progress toward the solution is maintained using merit functions.     

On Tightness and Weak Convergence in the Approximation of the Occupation Measure of Fractional Brownian Motion

In this paper we consider approximations of the occupation measure of the Fractional Brownian motion by means of some functionals defined on regularizations of the paths. In a previous article Berzin and León proved a cylindrical convergence to a Wiener process of conveniently rescaled functionals. Here we show the tightness of the approximation in the space of continuous functions endowed with the topology of uniform convergence on compact sets. This allows us to simplify the identification of the limit.     

Wavelet Approximation of Distributions with Bounded Variation Derivatives

Let BV ^r denote the space of distributions f such that the distributional derivatives D ^α f with |α|≤r exist as measures of bounded variation. This paper discusses estimates for wavelet coefficients of BV ^r distributions, direct (Jackson) and inverse (Bernstein) inequalities for n-term approximation of elements of BV ^r in the L _p spaces using compactly supported wavelets. In particular, optimal rates of approximation are established. Linear approximation in similar contexts is also considered for comparison. This research was supported by the 2003–2007 Academic Grant of Prof. P. Wojtaszczyk awarded by the Foundation for Polish Science. Part of this research was supported within the HASSIP framework.     

Hypercontractivity and Absolute Continuity of Gaussian Generalized Mehler Semigroups in Hilbert Spaces

In this paper, we prove the hypercontra,ctivity of a non-differentiable Gaussian generalized Mehler semigroup using direct probabilistic argumcents, This result implics the exponential convergence of the scmigroup at infinity. Under some additional hypotheses, we also） establish the absolute continuity of the semigroup with respect to its invariant mcasure.     

Absolute Continuity for the Occupation Times of Supper-Brownian Motion in a Super-Brownian Medium

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Infinite-Dimensional Quadrature and Approximation of Distributions

We study numerical integration of Lipschitz functionals on a Banach space by means of deterministic and randomized (Monte Carlo) algorithms. This quadrature problem is shown to be closely related to the problem of quantization and to the average Kolmogorov widths of the underlying probability measure. In addition to the general setting, we analyze, in particular, integration with respect to Gaussian measures and distributions of diffusion processes. We derive lower bounds for the worst case error of every algorithm in terms of its cost, and we present matching upper bounds, up to logarithms, and corresponding almost optimal algorithms. As auxiliary results, we determine the asymptotic behavior of quantization numbers and Kolmogorov widths for diffusion processes.      

Convergence of Algorithms in Optimization and Solutions of Nonlinear Equations

It is desirable that an algorithm in unconstrained optimization converges when the guessed initial position is anywhere in a large region containing a minimum point. Furthermore, it is useful to have a measure of the rate of convergence which can easily be computed at every point along a trajectory to a minimum point. The Lyapunov function method provides a powerful tool to study convergence of iterative equations for computing a minimum point of a nonlinear unconstrained function or a solution of a system of nonlinear equations. It is surprising that this popular and powerful tool in the study of dynamical systems is not used directly to analyze the convergence properties of algorithms in optimization. We describe the Lyapunov function method and demonstrate how it can be used to study convergence of algorithms in optimization and in solutions of nonlinear equations. We develop an index which can measure the rate of convergence at all points along a trajectory to a minimum point and not just at points in a small neighborhood of a minimum point. Furthermore this index can be computed when the calculations are being carried out.     

A Measure of Non-convexity in the Plane and the Minkowski Sum

In this paper a measure of non-convexity for a simple polygonal region in the plane is introduced. It is proved that for “not far from convex” regions this measure does not decrease under the Minkowski sum operation, and guarantees that the Minkowski sum has no “holes”.     

Statistical Convergence in Probability for a Sequence of Random Functions

In this paper, we introduce a new type of convergence for a sequence of random functions, namely, statistical convergence in probability, which is a natural generalization of convergence in probability. In this approach, we allow such a sequence to go far away from the limit point infinitely many times by presenting random deviations, provided that these deviations are negligible in some sense of measure. In this context, the set of values of a random function is considered as a probabilistic metric (PM) space of random variables, and some basic results are obtained using the tools of PM spaces.     

On Convergence of Chlodovsky and Chlodovsky–Kantorovich Polynomials in the Variation Seminorm

The aim of this paper is to study the variation detracting property and rate of approximation of the Chlodovsky and Chlodovsky–Kantorovich polynomials in the space of functions of bounded variation with respect to the variation seminorm.     

Convergence Rates of the Distributions of Error Variance Estimates in Linear Models

The best possible rate of convergence of the distributions of error variance estimates in linear models, based on the residual sum of squares, is obtained under weakest possible conditions.     

Absolute Continuity of a Two-Dimensional Magnetic Periodic Schrödinger Operator ith Potentials of the Type of Measure Derivative

A two-dimensional magnetic periodic Schrödinger operator with a variable metric is considered. An electric potential is assumed to be a distribution formally given by an expression , where d is a periodic signed measure with a locally finite variation. We also assume that the perturbation generated by the electric potential is strongly subject (in the sense of forms) to the free operator. Under this natural assumption, we prove that the spectrum of the Schrödinger operator is absolutely continuous. Bibliography: 15 titles.     

Strong Convergence of Iterative Algorithms for Variational Inequalities in Banach Spaces

Let C be a nonempty closed convex subset of a Banach space E with the dual E ^*, let T:C→E ^* be a Lipschitz continuous mapping and let S:C→C be a relatively nonexpansive mapping. In this paper, by employing the notion of generalized projection operator, we study the following variational inequality (for short, VI(T−f,C)): find x∈C such that

Convergence criterion and convergence ball of the Newton-type method in Banach space

The paper is concerned with the convergence problem of third-order Newton-type methods for finding zeros of nonlinear operations in Banach spaces. Under the hypothesis that the derivative of f satisfies the weak Lipschitz condition with L-average, the convergence criterion and convergence ball are given. Furthermore, some corollaries are obtained by applying the main results to some special functions L.     

Weak Convergence of an Iterative Method for Pseudomonotone Variational Inequalities and Fixed-Point Problems

We consider an iterative scheme for finding a common element of the set of solutions of a pseudomonotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of N nonexpansive mappings. The proposed iterative method combines two well-known schemes: extragradient and approximate proximal methods. We derive a necessary and sufficient condition for weak convergence of the sequences generated by the proposed scheme.     

Hölder Continuity of the Saddle Point Set for Real-Valued Functions

This paper is concerned with Hölder continuity of the solution to a saddle point problem. Some new su?cient conditions for the uniqueness and Hölder continuity of the solution for a perturbed saddle point problem are established. Applications of the result on Hölder continuity of the solution for perturbed constrained optimization problems are presented under mild conditions. Examples are given to illustrate the obtained results.     

Convergence Properties of Modified and Partially-Augmented Lagrangian Methods for Mathematical Programs with Complementarity Constraints

We present new convergence properties of partially augmented Lagrangian methods for mathematical programs with complementarity constraints (MPCC). Four modified partially augmented Lagrangian methods for MPCC based on different algorithmic strategies are proposed and analyzed. We show that the convergence of the proposed methods to a B-stationary point of MPCC can be ensured without requiring the boundedness of the multipliers.     

RETRACTED ARTICLE: Convergence of Weighted Sums for Arrays of Negatively Dependent Random Variables and Its Applications

We discuss the complete convergence of weighted sums for arrays of rowwise negatively dependent random variables (ND r.v.’s) to linear processes. As an application, we obtain the complete convergence of linear processes based on ND r.v.’s which extends the result of Li et al. (Stat. Probab. Lett. 14:111–114, 1992), including the results of Baum and Katz (Trans. Am. Math. Soc. 120:108–123, 1965), from the i.i.d. case to a negatively dependent (ND) setting. We complement the results of Ahmed et al. (Stat. Probab. Lett. 58:185–194, 2002) and confirm their conjecture on linear processes in the ND case.     

Weak Convergence Results for Multiple Generations of a Branching Process

We establish limit theorems involving weak convergence of multiple generations of critical and supercritical branching processes. These results arise naturally when dealing with the joint asymptotic behavior of functionals defined in terms of several generations of such processes. Applications of our main result include a functional central limit theorem (CLT), a Darling–Erdös result, and an extremal process result. The limiting process for our functional CLT is an infinite dimensional Brownian motion with sample paths in the infinite product space (C ₀[0,1])^∞, with the product topology, or in Banach subspaces of (C ₀[0,1])^∞ determined by norms related to the distribution of the population size of the branching process. As an application of this CLT we obtain a central limit theorem for ratios of weighted sums of generations of a branching processes, and also to various maximums of these generations. The Darling–Erdös result and the application to extremal distributions also include infinite-dimensional limit laws. Some branching process examples where the CLT fails are also included.