On Logarithmic Smoothing of the Maximum Function

We consider the maximum function f resulting from a finite number of smooth functions. The logarithmic barrier function of the epigraph of f gives rise to a smooth approximation g of f itself, where >0 denotes the approximation parameter. The one-parametric family g converges – relative to a compact subset – uniformly to the function f as tends to zero. Under nondegeneracy assumptions we show that the stationary points of g and f correspond to each other, and that their respective Morse indices coincide. The latter correspondence is obtained by establishing smooth curves x() of stationary points for g , where each x() converges to the corresponding stationary point of f as tends to zero. In case of a strongly unique local minimizer, we show that the nondegeneracy assumption may be relaxed in order to obtain a smooth curve 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.     

Range of the posterior probability of an interval for priors with unimodality preserving contaminations

Range of the posterior probability of an interval over the -contamination class ={=(1–)₀+q:qQ} is derived. Here, ₀ is the elicited prior which is assumed unimodal, is the amount of uncertainty in ₀, andQ is the set of all probability densitiesq for which =(1–)₀+q is unimodal with the same mode as that of ₀. We show that the sup (resp. inf) of the posterior probability of an interval is attained by a prior which is equal to (1–)₀ except in one interval (resp. two disjoint intervals) where it is constant.     

On the projected subgradient method for nonsmooth convex optimization in a Hilbert space

We consider the method for constrained convex optimization in a Hilbert space, consisting of a step in the direction opposite to an _k -subgradient of the objective at a current iterate, followed by an orthogonal projection onto the feasible set. The normalized stepsizes _k are exogenously given, satisfying _{k=0 k} = , _{k=0 k} ² < , and _k is chosen so that _{k k} for some > 0. We prove that the sequence generated in this way is weakly convergent to a minimizer if the problem has solutions, and is unbounded otherwise. Among the features of our convergence analysis, we mention that it covers the nonsmooth case, in the sense that we make no assumption of differentiability off, and much less of Lipschitz continuity of its gradient. Also, we prove weak convergence of the whole sequence, rather than just boundedness of the sequence and optimality of its weak accumulation points, thus improving over all previously known convergence results. We present also convergence rate results. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Research of this author was partially supported by CNPq grant nos. 301280/86 and 300734/95-6.     

An algebraic approach to the vector ε-algorithm

The vector -algorithm is obtained from the scalar -algorithm by taking the pseudo-inverse of a vector instead of the inverse of a scalar. Thus the vector -algorithm is known only through its rules contrarily to the scalar -algorithm and some other extrapolation algorithms.The aim of this paper is to provide an algebraic approach to the vector -algorithm.     

Fuzzy Principles and Characterization of Trustworthiness

We show that (a) the basic principles of subdifferential calculus (various local and global fuzzy principles, multidirectional mean value theorem, extremal principle) which are shown to be equivalent for -subdifferentials are in fact equivalent for any subdifferential and (b) for -versions of -subdifferentials these principles turn out to be also equivalent to the properties of the space to be a trustworthy or a subdifferentiability space (with respect to a given subdifferential).     

Properties of the Mappings That Are Close to the Harmonic Mappings. II

We continue studying the mappings that are close to the harmonic mappings (-quasiharmonic mappings with small). This study originates with the previous articles of the author. The results of the article include a theorem on connection between the notion of -quasiharmonic mapping and the solutions to Beltrami systems, an analog to the arithmetic mean property of harmonic functions for -quasiharmonic mappings, a theorem on stability in the Poisson formula for harmonic mappings in the ball, and a theorem on the local smoothing of -quasiharmonic mappings with small which preserves proximity to the harmonic mappings.     

Singularly perturbed functional-differential inclusions

Differential inclusions of a retarded type with a small real parameter >0 in part of the derivatives are considered. We prove upper semicontinuity of the map set of solutions at =0⁺ inC[0, 1]×(L ²(0, 1)–weak) topology. In case of constant delay lower semicontinuity inC[0, 1]×(L ¹(0, 1)–strong) is shown.     

Global optimization of univariate Lipschitz functions: II. New algorithms and computational comparison

We consider the following global optimization problems for a Lipschitz functionf implicitly defined on an interval [a, b]. Problem P: find a globally-optimal value off and a corresponding point; Problem Q: find a set of disjoint subintervals of [a, b] containing only points with a globally-optimal value and the union of which contains all globally optimal points. A two-phase algorithm is proposed for Problem P. In phase I, this algorithm obtains rapidly a solution which is often globally-optimal. Moreover, a sufficient condition onf for this to be the case is given. In phase II, the algorithm proves the-optimality of the solution obtained in phase I or finds a sequence of points of increasing value containing one with a globally-optimal value. The new algorithm is empirically compared (on twenty problems from the literature) with a best possible algorithm (for which the optimal value is assumed to be known), with a passive algorithm and with the algorithms of Evtushenko, Galperin, Shen and Zhu, Piyavskii, Timonov and Schoen. For small, the new algorithm requires only a few percent more function evaluations than the best possible one. An extended version of Piyavskii's algorithm is proposed for problem Q. A sufficient condition onf is given for the globally optimal points to be in one-to-one correspondance with the obtained intervals. This result is achieved for all twenty test problems.The research of the authors has been supported by AFOSR grants 0271 and 0066 to Rutgers University. Research of the second author has been also supported by NSERC grant GP0036426, FCAR grant 89EQ4144 and partially by AFOSR grant 0066. We thank Nicole Paradis for her help in drawing the figures.     

On the Korn Inequalities on Thin Periodic Frames

Korn-type inequalities for thin periodic structures of period and width h() with h() 0 are presented. Periodic meshes, three-dimensional road structures, and three-dimensional box structures are considered. A particular attention is paid to structures with the so-called critical width when 0$$ " align="middle" border="0"> .     

Inverse Problems in the Theory of Singular Perturbations

First, in joint work with S. Bodine of the University of Puget Sound, Tacoma, Washington, USA, we consider the second-order differential equation ² y'=(1+² (x, ))y with a small parameter , where is analytic and even with respect to . It is well known that it has two formal solutions of the form y^±(x,)=e^±x/h^±(x,), where h^±(x,) is a formal series in powers of whose coefficients are functions of x. It has been shown that one (resp. both) of these solutions are 1-summable in certain directions if satisfies certain conditions, in particular concerning its x-domain. We show that these conditions are essentially necessary for 1-summability of one (resp. both) of the above formal solutions. In the proof, we solve a certain inverse problem: constructing a differential equation corresponding to a certain Stokes phenomenon. The second part of the paper presents joint work with Augustin Fruchard of the University of La Rochelle, France, concerning inverse problems for the general (analytic) linear equations ^r y' = A(x,) y in the neighborhood of a nonturning point and for second-order (analytic) equations y' - 2xy'-g(x,) y=0 exhibiting resonance in the sense of Ackerberg-O'Malley, i.e., satisfying the Matkowsky condition: there exists a nontrivial formal solution such that the coefficients have no poles at x=0.     

The ε c -Product of a Schwartz b-Space by a Quotient Banach Space and Applications

We define the -product of a -space by a quotient Banach space. We give conditions under which this -product will be monic. Finally, we define the _c -product of a Schwartz b-space by a quotient Banach space and we give some examples of applications.     

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.     

Regularization of Nonlinear Ill-Posed Variational Inequalities and Convergence Rates

Let H be a Hilbert space and K be a nonempty closed convex subset of H. For f H, we consider the (ill-posed) problem of finding u K for which 0 for all v K, where A : H H is a monotone (not necessarily linear) operator. We study the approximation of the solutions of the variational inequality by using the following perturbed variational inequality: for f H, f – f , find u, K for which 0 for all v K, where , , and are positive parameters, and K, a perturbation of the set K, is a nonempty closed convex set in H. We establish convergence and a rate O(1 / 3) of convergence of the solutions of the regularized variational inequalities to a solution of the original variational inequality using the Mosco approximation of closed convex sets, where A is a weakly differentiable inverse-strongly-monotone operator.     

Almost flat locally finite coverings of the sphere

For any subset A of the unit sphere of a Banach space X and for [0,2) the notion of -flatness is introduced as a measure of non-flatness of A. For any positive , construction of locally finite tilings of the unit sphere by -flat sets is carried out under suitable -renormings of X in a quite general context; moreover, a characterization of spaces having separable dual is provided in terms of the existence of such tilings. Finally, relationships between the possibility of getting such tilings of the unit sphere in the given norm and smoothness properties of the norm are discussed.     

Quantitative stability of variational systems: III.ε-approximate solutions

We prove that the-optimal solutions of convex optimization problems are Lipschitz continuous with respect to data perturbations when these are measured in terms of the epi-distance. A similar property is obtained for the distance between the level sets of extended real valued functions. We also show that these properties imply that the-subgradient mapping is Lipschitz continuous.Research supported in part by the National Science Foundation and the Air Force Office of Scientific Research.     

On the complexity of approximating a KKT point of quadratic programming

We present a potential reduction algorithm to approximate a Karush—Kuhn—Tucker (KKT) point of general quadratic programming (QP). We show that the algorithm is a fully polynomial-time approximation scheme, and its running-time dependency on accuracy (0, 1) is O((l/) log(l/) log(log(l/))), compared to the previously best-known result O((l/)²). Furthermore, the limit of the KKT point satisfies the second-order necessary optimality condition of being a local minimizer. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Research support in part by NSF grants DDM-9207347 and DMI-9522507, and the Iowa Business School Summer Grant.     

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.     

Chebyshev Solution of the Nearly-Singular One-Dimensional Helmholtz Equation and Related Singular Perturbation Equations: Multiple Scale Series and the Boundary Layer Rule-of-Thumb

The one-dimensional Helmholtz equation, ² u _xx –u=f(x), arises in many applications, often as a component of three-dimensional fluids codes. Unfortunately, it is difficult to solve for 1 because the homogeneous solutions are exp(±x/), which have boundary layers of thickness O(1/). By analyzing the asymptotic Chebyshev coefficients of exponentials, we rederive the Orszag–Israeli rule [16] that Chebyshev polynomials are needed to obtain an accuracy of 1% or better for the homogeneous solutions. (Interestingly, this is identical with the boundary layer rule-of-thumb in [5], which was derived for singular functions like tanh([x–1]/).) Two strategies for small are described. The first is the method of multiple scales, which is very general, and applies to variable coefficient differential equations, too. The second, when f(x) is a polynomial, is to compute an exact particular integral of the Helmholtz equation as a polynomial of the same degree in the form of a Chebyshev series by solving triangular pentadiagonal systems. This can be combined with the analytic homogeneous solutions to synthesize the general solution. However, the multiple scales method is more efficient than the Chebyshev algorithm when is very, very tiny.     

On the Pellian Equation

Let (d) be the least solution of the Pellian equation x ²-dy ² = 1. It is proved that there exists a sequence of values of d having a positive density and such that (d) > d ^2-, where is an arbitrary positive constant. Bibliogrhaphy: 7 titles.