Convergence Analysis of Perturbed Feasible Descent Methods

We develop a general approach to convergence analysis of feasible descent methods in the presence of perturbations. The important novel feature of our analysis is that perturbations need not tend to zero in the limit. In that case, standard convergence analysis techniques are not applicable. Therefore, a new approach is needed. We show that, in the presence of perturbations, a certain -approximate solution can be obtained, where depends linearly on the level of perturbations. Applications to the gradient projection, proximal minimization, extragradient and incremental gradient algorithms are described.     

Interface Fluctuations and Couplings in the D=1 Ginzburg–Landau Equation with Noise

We consider a Ginzburg–Landau equation in the interval [–^–, ^–], >0, 1, with Neumann boundary conditions, perturbed by an additive white noise of strength We prove that if the initial datum is close to an "instanton" then, in the limit 0⁺, the solution stays close to some instanton for times that may grow as fast as any inverse power of , as long as the center of the instanton is far from the endpoints of the interval. We prove that the center of the instanton, suitably normalized, converges to a Brownian motion. Moreover, given any two initial data, each one close to an instanton, we construct a coupling of the corresponding processes so that in the limit 0⁺ the time of success of the coupling (suitably normalized) converges in law to the first encounter of two Brownian paths starting from the centers of the instantons that approximate the initial data.     

Homogenization of Some Elasticity Problems with Rapidly Oscillating Convex Constraints on Displacements

The problem of homogenization is considered for an elastic body occupying a perforated domain = obtained from a fixed domain and an -contraction of a 1-periodic domain .     

Convergence of the Gradient Projection Method for Generalized Convex Minimization

This paper develops convergence theory of the gradient projection method by Calamai and Moré (Math. Programming, vol. 39, 93–116, 1987) which, for minimizing a continuously differentiable optimization problem min{f(x) : x } where is a nonempty closed convex set, generates a sequence x_k+1 = P(x_k – _k f(x_k)) where the stepsize _k > 0 is chosen suitably. It is shown that, when f(x) is a pseudo-convex (quasi-convex) function, this method has strong convergence results: either x_k x^* and x^* is a minimizer (stationary point); or x_k arg min{f(x) : x } = , and f(x_k) inf{f(x) : x }.     

Enlargement of Monotone Operators with Applications to Variational Inequalities

Given a point-to-set operator T, we introduce the operator T defined as T(x)= {u: u – v, x – y – for all y Rn, v T(y)}. When T is maximal monotone T inherits most properties of the -subdifferential, e.g. it is bounded on bounded sets, T(x) contains the image through T of a sufficiently small ball around x, etc. We prove these and other relevant properties of T, and apply it to generate an inexact proximal point method with generalized distances for variational inequalities, whose subproblems consist of solving problems of the form 0 H(x), while the subproblems of the exact method are of the form 0 H(x). If k is the coefficient used in the kth iteration and the k's are summable, then the sequence generated by the inexact algorithm is still convergent to a solution of the original problem. If the original operator is well behaved enough, then the solution set of each subproblem contains a ball around the exact solution, and so each subproblem can be finitely solved.     

A Simplified Convergence Proof for the Cone Partitioning Algorithm

We present a new convergence result for the cone partitioning algorithm with a pure -subdivision strategy, for the minimization of a quasiconcave function over a polytope. It is shown that the algorithm is finite when -optimal solution with > 0 are looked for, and that any cluster point of the points generated by the algorithm is an optimal solution in the case = 0. This result improves on the one given previously by the authors, its proof is simpler and relies more directly on a new class of hyperplanes and its associated simplicial lower bound.     

Boundary-layer averaging for standard systems with lag

Thekth-order asymptotic solution of a standard system with lag is constructed along trajectories calculated according to the averaging scheme of A. N. Filatov. If the perturbation parameter 1, then the use of the step method for finding the solution is connected with cumbersome calculations because the number of required steps is inversely proportional to . We suggest another approach in which the step method is used onlyk times fort [0,k] and justify the asymptotic method.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10 pp. 1362–1368, October, 1994.     

On the complex zeros of solutions of linear differential equations

Summary A classical result (see R.Nevanlinna, Acta Math.,58 (1932), p. 345) states that for a second-order linear differential equation, w + P(z) w + Q(z) w=0, where P(z) and Q(z) are polynomials, there exist finitely many rays, arg z=_j, for j=1,..., m, such that for any solution w=f(z) 0 and any > 0, all but finitely many zeros off lie in the union of the sectors ¦ arg z - _j¦ < for j=1,..., m. In this paper, we give a complete answer to the question of determining when the same result holds for equations of arbitrary order having polynomial coefficients. We prove that for any such equation, one of the following two properties must hold: (a) for any ray, arg z=, and any > 0, there is a solution f 0 of the equation having infinitely many zeros in the sector ¦arg z - ¦ <, or (b) there exist finitely many rays, arg z=_j, for j= 1,..., m, such that for any >0, all but finitely many zeros of any solution f 0 must lie in the union of the sectors ¦ arg z - _j¦ < for j=1, ..., m. In addition, our method of proof provides an effective procedure for determining which of the two possibilities holds for a given equation, and in the case when (b) holds, our method will produce the rays, arg z=_j. We emphasize that our result applies to all equations having polynomial coefficients, without exception. In addition, we mention that if the coefficients are only assumed to be rational functions, our results will still give precise information on the possible location of the bulk of the zeros of any solution.This research was supported in part by the National Science Foundation (DMS-84-20561 and DMS-87-21813).     

Asymptotic Behavior of Higher-Order Perturbations: Scaling Functions of the O(n)-Symmetric φ4-Theory in the (4−ε)-Expansion

We use an instantonic approach to calculate the asymptotic behavior of higher orders of the (4–)-expansion for the scaling function of the pair correlator of the O(n)-symmetric ⁴-theory in the minimal subtraction scheme. Our results differ substantially from the known exact expression for the ³ order of the expansion of the scaling function in the small- domain.     

Shift-coupling in continuous time

Summary The result linking shift-coupling to time-average total variation convergence and to the invariant -field is extended to continuous time and an analogous result established linking -couplings to smooth total variation convergence and to a smooth tail -field. Shift- and -coupling inequalities are presented.