Cone Adaptation Strategies for a Finite and Exact Cutting Plane Algorithm for Concave Minimization

In this paper we are concerned with pure cutting plane algorithms for concave minimization. One of the most common types of cutting planes for performing the cutting operation in such algorithm is the concavity cut. However, it is still unknown whether the finite convergence of a cutting plane algorithm can be enforced by the concavity cut itself or not. Furthermore, computational experiments have shown that concavity cuts tend to become shallower with increasing iteration. To overcome these problems we recently proposed a procedure, called cone adaptation, which deepens concavity cuts in such a way that the resulting cuts have at least a certain depth with 0, where is independent of the respective iteration, which enforces the finite convergence of the cutting plane algorithm. However, a crucial element of our proof that these cuts have a depth of at least was that we had to confine ourselves to -global optimal solutions, where is a prescribed strictly positive constant. In this paper we examine possible ways to ensure the finite convergence of a pure cutting plane algorithm for the case where = 0.     

A Multivariate Global Optimization Using Linear Bounding Functions

Recently linear bounding functions (LBFs) were proposed and used to find -global minima. This paper presents an LBF-based algorithm for multivariate global optimization problems. The algorithm consists of three phases. In the global phase, big subregions not containing a solution are quickly eliminated and those which possibly contain the solution are detected. An efficient scheme for the local phase is developed using our previous local minimization algorithm, which is globally convergent with superlinear/quadratic rate and does not require evaluation of gradients and Hessian matrices. To ensure that the found minimizers are indeed the global solutions or save computation effort, a third phase called the verification phase is often needed. Under adequate conditions the algorithm finds the -global solution(s) within finite steps. Numerical testing results illustrate how the algorithm works, and demonstrate its potential and feasibility.     

Shift invariant spaces inL 2

If is a complex, separable Hilbert space, letL ² () denote theL ²-space of functions defined on the unit circle and having values in . The bilateral shift onL ²() is the operator (U f)()=f(). A Hilbert spaceH iscontractively contained in the Hilbert spaceK ifHK and the inclusion mapH–K is a contraction. We describe the structure of those Hilbert spaces, contractively contained inL ²(), that are carried into themselves contractively byU . We also do this for the subcase of those spaces which are carried into themselves unitarily byU .     

Lower semicontinuity of attractors of gradient systems and applications

Summary For 0₀, let T(t), t0, be a family of semigroups on a Banach space X with local attractors A. Under the assumptions that T₀(t) is a gradient system with hyperbolic equilibria and T(t) converges to T₀(t) in an appropriate sense, it is shown that the attractors {A, 0₀} are lower-semicontinuous at zero. Applications are given to ordinary and functional differential equations, parabolic partial differential equations and their space and time discretizations. We also give an estimate of the Hausdorff distance between A and A₀, in some examples.Research supported by U.S. Army Research Office DAAL-03-86-K-0074 and the National Science Foundation DMS-8507056.     

Estimates of the asphericity of sections of convex bodies

We define the function (n, k) to be the infimum of all such that any bounded centrally symmetric convex body inR ⁿ possesses an -asphericalk-dimensional central section. It is proved that (3, 2)=2–1 and (n, n-1)n-1-1. Several related functions are defined and their values on the pairs (n, n-1) are estimated.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 76–79.     

Inverse power law potentials in rectangular configurations

Summary Calculations based on a (distance)^– intermolecular potential (>3) enable study of the effects on adsorption of the geometry of the solid. This paper gives the closed form solution for the adsorptive potential about a homogeneous solid rectangular corner; and, through systematic superposition, closed form solutions for the following configurations also: the rectangular corner of a cavity; laminae and rectangular cracks occupying a quarter plane; semi-infinite rectangular prisms and prismatic cavities; rectangular parallelepipeds and brick-shaped cavities. These various results are developed in detail for the cases =6 and =4. The paradox that potentials for >3 seem to be obtainable more readily than Newtonian potentials (=1) is explained by the existence only for >3 of simple fundamental solutions for infinite homogeneous solid configurations.

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Zusammenfassung Berechnungen, denen ein intermolekulares Potential der Form (Abstand)^– (>3) zugrunde gelegt ist, ermöglichen eine Untersuchung von Effekten der Adsorption auf die Geometrie des Festkörpers. Die vorliegende Arbeit gibt die Lösung in geschlossener Form für das Adsorptionspotential um eine feste, homogene, rechtwinklige Ecke an. Ausserdem werden durch systematische Superposition Lösungen in geschlossener Form für die folgenden Konfigurationen angegeben: die rechtwinklige Innenecke einer Mulde; viertelunendliche, ebene Platten und rechteckige Spalten; halbunendliche, reckteckige Prismen und prismatische Mulden; Quader und quaderförmige Höhlen. Diese Ergebnisse sind ausführlich dargestellt für die Fälle =4. Das Paradoxon. dass Potentiale mit >3 scheinbar leichter zugänglich sind als das Gravitationspotential (=1), wird dadurch erklärt, dass nur für >3 einfache Grundlösungen für unendliche, homogene Festköperkonfigurationen existieren.

     

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.     

Homogenization and concentrated capacity for the heat equation with non-linear variational data in reticular almost disconnected structures and applications to visual transduction

Out of a right, circular cylinder of height H and cross-section a disc of radius R+ one removes a stack of nH/ parallel, equi-spaced cylinders C_j,j=1,2,...,n, each of radius R and height . Here , are fixed positive numbers and is a positive parameter to be allowed to go to zero. The union of the C_j almost fills in the sense that any two contiguous cylinders C_j are at a mutual distance of the order of and that the outer shell, i.e., the gap S=-_o has thickness of the order of (_o is obtained from by formally setting =0). The cylinder from which the C_j are removed, is an almost disconnected structure, it is denoted by , and it arises in the mathematical theory of phototransduction.For each >0 we consider the heat equation in the almost disconnected structure , for the unknown function u, with variational boundary data on the faces of the removed cylinders C_j. The limit of this family of problems as 0 is computed by concentrating heat capacity and diffusivity on the outer shell, and by homogenizing the u within the limiting cylinder _o.It is shown that the limiting problem consists of an interior diffusion in _o and a boundary diffusion on the lateral boundary S of _o. The interior diffusion is governed by the 2-dimensional heat equation in _o, for an interior limiting function u. The boundary diffusion is governed by the Laplace–Beltrami heat equation on S, for a boundary limiting function u_S. Moreover the exterior flux of the interior limit u provides the source term for the boundary diffusion on S. Finally the interior limit u, computed on S in the sense of the traces, coincides with the boundary limit u_S. As a consequence of the geometry of , local arguments do not suffice to prove convergence in _o, and also we have to take into account the behavior of the solution in S. A key, novel idea consists in extending equi-bounded and equi-Hölder continuous functions in -dependent domains, into equi-bounded and equi-Hölder continuous functions in the whole ^N, by means of the Kirzbraun–Pucci extension technique.The biological origin of this problem is traced, and its application to signal transduction in the retina rod cells of vertebrates is discussed. Mathematics Subject Classification (2000) 35B27, 35K50, 92C37     

Characterization of Dirichlet Forms by their Excessive and Co–Excessive Elements

It is proved that if S, T are two elliptic Dirichlet operators on an ordered Hilbert space such that the excessive (resp. coexcessive) elements with respect to S and T are the same then there exists > 0 with T = S. Particularly if , are two elliptic Dirichlet forms on L² ( ) having the same domain of definition and the same -excessive (resp. -coexcessive) elements for any > 0 then = .     

Spectral parameter asymptotics of the Weyl solutions of Sturm-Liouville equations

In the paper one investigates the dependence of Weyl's solution ,)=c(,)+n()s(,) of the Sturm-Liouville equation y+q()y=²y on the spectral parameter . Under the condition that the potential q is bounded from below and q()exp(c₀+c[in1 ¦¦), it is proved for {ie217-01} for any positive values and A. If q()>1 and {ie217-02} for all >0, then in the semiplane >0 the Weyl solution (, ) is obtained from the Weyl solution (,x) is obtained from the Weyl solution e^ix with zero potential, with the aid of a generalization of B. Ya Levin's transformation operators.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 184–206, 1989.I express my sincere gratitude to L. A. Pastur and I. V. Ostrovskii for valuable advice and discussions.     

Gegen Loten abgeschlossene Untergruppen metrischer Vektorräume

The purpose of this paper is to describe for any finite dimensional anisotropic metric vector space V=V(K,f) over a field K all those subgroups U of the additive group of V satisfying the condition: If f(a,b)=0 and a+b U then a,b U, for all a,b V.     

Homogenization of a Singularly Perturbed Parabolic Problem in a Thick Periodic Junction of the Type 3:2:1

We prove a convergence theorem and obtain asymptotic (as 0) estimates for a solution of a parabolic initial boundary-value problem in a junction that consists of a domain ₀ and a large number N ² of -periodically located thin cylinders whose thickness is of order = O(N ^–1).     

Spectral Portraits of the Orr-Sommerfeld Operator with Large Reynolds Numbers

The model problem -i y'+q(x)y= y, y(-1)=y(1)=0 is associated with the Orr-Sommerfeld operator well-known in hydrodynamics. Here is the spectral parameter, is the small parameter which is proportional to the viscosity of the liquid and to the reciprocal of the Reynolds number, and q(x) is the velocity of the stationary flow of the liquid in the channel |x| 1. We study the behavior of the spectrum of the corresponding model operator as to 0 with linear, quadratic, and monotonic analytic functions. We show that the sets of accumulation points of the spectra (the limit spectral graphs) of the model and corresponding Orr-Sommerfeld operators coincide just as the main terms of the eigenvalue counting functions along the curves of the graphs do.     

Retardation trace in tension with subsequent compression in the transverse direction

The local strains theory is employed to determine the strain components ₁ and ₂ in five-dimensional space in the case of tension with subsequent compression at nonlinearities n=3 and n=5. The relations between the vectors and S and the tangents to the loading and deformation trajectories and between the moduli of the strain for complex and simple loading are considered.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 597–602, July–August, 1969.     

How to approximate the solutions of certain free boundary problems for the Laplace equation by using the contraction principle

We show how the free boundary of an ideal fluid, subject to a generalized Bernoulli condition, can (under appropriate circumstances) be approximated. Our method is based on a class of free-boundary perturbation operatorsT , 0<<1, which are all contracting relative to a suitable norm and class of boundaries, and whose fixed points converge to the desired free boundary solution as 0+.

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Zusammenfassung Wir zeigen, wie der freie Rand einer idealen Flüssigkeit, welcher einer verallgemeinerten Bernoulli-Bedingung genügt, unter geeigneten Umständen approximiert werden kann. Unsere Methode stützt sich auf eine Klasse freier RandperturbationsoperatorenT , 0<<1, welche relativ zu einer geeigneten Norm und Ränderklasse kontrahierend sind und deren Fixpunkte gegen die gewünschte Lösung der freien Randaufgabe mit 0+ konvergieren.

     

On Localization and Stabilization for Factorization Systems

If (, M)is a factorization system on a category C, we define new classes of maps as follows: a map f:AB is in if each of its pullbacks lies in (that is, if it is stably in ), and is in M ^* if some pullback of it along an effective descent map lies in M(that is, if it is locally in M). We find necessary and sufficient conditions for (, M ^*) to be another factorization system, and show that a number of interesting factorization systems arise in this way. We further make the connexion with Galois theory, where M ^*is the class of coverings; and include self-contained modern accounts of factorization systems, descent theory, and Galois theory.     

Pointwise displacement errors in linear shell theory resulting from errors in the stress-strain relations

It is rigorously proved that relative errors of order in the stress-strain relations of linear shell theory result in relative pointwise errors in the solution displacement field of order .

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Zusammenfassung Für die Theorie dünner Schalen wird bewiesen, daß ein relativer Fehler der Größe in den Spannungs-Dehnungs-Beziehungen einen relativen lokalen Fehler der Größe in der Lösung für das Verschiebungsfeld erzeugt.

     

Periodic solutions of singularly perturbed delay equations

We consider a general class of singularly perturbed delay differential systems depending on a singular parameter and another parameter . For =0, the equation defines a mapT which undergoes a generic period doubling at =0. If the bifurcation is supercritical (subcritical), these period two points define a stable period two square wave (unstable period two pulse wave). We give conditions on the vector field such that there is a sectorS in the , plane such that there is a unique periodic orbit if the parameters are inS, the orbit is stable (unstable) if the period doubling bifurcation is supercritical (subcritical) and approaches the square (pulse) wave as 0.Partially supported by NSF and DARPA.     

Decomposition based and branch and bound global optimization approaches for the phase equilibrium problem

An increasingly popular approach when solving the phase and chemical equilibrium problem is to pose it as an optimization problem. However, difficulties are encountered due to the highly nonlinear nature of the models used to represent the behavior of the fluids, and because of the existence of multiple local solutions. This work shows how it is possible to guarantee -global solutions for a certain important class of the phase and chemical equilibrium problem, namely when the liquid phase can be modeled using neither the Non-Random Two-Liquid (NRTL) equation, or the UNIversal QUAsi Chemical (UNIQUAC) equation. Ideal vapor phases are easily incorporated into the global optimization framework. A numberof interesting properties are described which drastically alter the structure of the respective problems. For the NRTL equation, it is shown that the formulation can be converted into a biconvex optimization problem. The GOP algorithm of Floudas and Visweswaran [8, 9] can then be used to obtain -global solutions in this case. For the UNIQUAC equation, the new properties show how the objective function can be transformed into the difference of two convex functions (i.e. a D.C. programming problem is obtained), where the concave portion is separable. A branch and bound algorithm based on that of Falk and Soland [6] is used to guarantee convergence to an -global solution. Examples are presented which demonstrate the performance of both algorithms.     

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.