Humboldt Universität zu Berlin  Naturwissenschaftliche Fakultät II  Institut für Mathematik WWW-Adresse: http://www2.mathematik.hu-berlin.de/publ/pre/1996

Preprints: Veröffentlichungsliste 1996

PREPRINT 96-1

Schied, Alexander

Sample path large deviations for super-Brownian motion

PREPRINT 96-2

Henrion, René; Römisch, Werner

Metric regularity and quantitative stability in stochastic programs with probabilistic constraints

Abstract:
Necessary and sufficient conditions for metric regularity of (several joint) probabilistic con- straints are derived using results from nonsmooth analysis. The conditions apply to fairly general nonconvex, nonsmooth probabilistic constraints and extend earlier work in this direction. Further, a verifiable sufficient condition for quadratic growth of the objective function in a more specific convex stochastic program is indicated and applied in order to obtain a new result on quantitative stability of solution sets when the underlying probability distribution is subjected to perturbations. This is used to establish a large deviation estimate for solution sets when the probability measure is replaced by empirical ones.

PREPRINT 96-3

Wolff, Michael

A global L*-gradient estimate on weak solutions to nonlinear stationary Navier- Stokes equations under mixed boundary conditions

PREPRINT 96-4

März, Roswitha; Tischendorf, Caren

Recent results in solving index 2 differential-algebraic equations in circuit simulation

to appear in SIAM J. Sci. Stat. Comput., 18 (1), 1997.

Abstract:
In electric circuit simulation the charge oriented modified nodal analysis may lead to highly nonlinear DAEs with low smoothness properties. They may have index 2 but they do not belong to the class of Hessenberg form systems that are well understood.
In the present paper, on the background of a detailed analysis of the resulting structure, it is shown that charge oriented modified nodal analysis yields the same index as the classical modified nodal analysis.
Moreover, for index 2 DAEs in the charge oriented case, a further careful analysis with respect to solvability, linearization and numerical integration is given.

Keywords: Differential-algebraic equations, index 2, circuit simulation, IVP, numerical integration, BDF, defect correction.

PREPRINT 96-5

Gomez Bofill, Walter

Vector optimization: Singularities, regularizations

Abstract:
We discuss three scalarizations of the multiobjective optimization from the point of view of the parametric optimization. We analize three important aspects:

 
1. What kind of singularities may appear in the different parametrizations
2. Regularizations in the sense of Jongen, Jonker and Twilt, and in the sense of Kojima and Hirabayashi.
3. The Mangasarian- Fromovitz Constraint Qualification for the first parametrization.

PREPRINT 96-6

Dentcheva, D.; Möller, A.; Reeh, P.; Römisch, W.; Schultz, R.; Schwarzbach, G.; Thomas, J.

Optimale Blockauswahl bei der Kraftwerkseinsatzplanung

Abstract:
The paper addresses the unit commitment problem in power plant operation planning. For a real power system comprising coal and gas fired thermal as well as pumped storage hydro plants a large-scale mixed integer optimization model for unit commitments is developed. Then primal and dual approaches to solving the optimization problem are presented and results of test runs are reported.

PREPRINT 96-7

Fandom, Noubiap R.

On Modifications of the Standard Embedding in Nonlinear Optimization

Abstract:
This paper deals with pathfollowing methods in nonlinear optimization. We study the so- called "standard embedding" and show its limits. Then, we modify this embedding from several points of view and obtain modified standard embeddings having some advantages. Singularity theory developed by Jongen-Jonker-Twilt plays a great role in our investigation. In some cases, we have to jump from one connected component to another one in the set of local minimizers and in the set of generalized critical points, respectively. In the worst case, we have to find all connected components and that is still an open problem. Computationary results are presented.

PREPRINT 96-8

Schied, Alexander

Geometric aspects of Fleming-Viot and superprocesses

Abstract:
The main purpose of this paper is to show that the intrinsic metric of the Fleming-Viot process is given by an "angular distance" on the space of probability measures. It turns out that it is closely related to the branching structure of a continuous superprocess, which itself induces the Kakutani-Hellinger distance. The corresponding geometries are studied in some detail. In particular, representation formulae for geodesics and arc length functionals are obtained. As an application, a functional limit theorem for super-Brownian motion conditioned to local extinction is proved.

PREPRINT 96-9

Schied, Alexander

Moderate deviations and functional LIL for super Brownian motion

PREPRINT 96-10

Bank, B.; Giusti, M.; Heintz, J.; Mandel, R.; Mbakop, G. M.

Polar Varieties and Efficient Real Equation Solving: The Hypersurface Case

Abstract
The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais, Morgenstern, Pardo can be applied to a case of real polynomial equation solving. Our main result concerns the problem of finding one representative point for each connected component of a real bounded smooth hypersurface.
The algorithm in yields a method for symbolically solving a zero-dimensional polynomial equation system in the affine (and toric) case. Its main feature is the use of adapted data structure: Arithmetical networks and straight-line programs. The algorithm solves any affine zero-dimensional equation system in non-uniform sequential time that is polynomial in the length of the input description and an adequately defined affine degree of the equation system.
Replacing the affine degree of the equation system by a suitably defined real degree of certain polar varieties associated to the input equation, which describes the hypersurface under consideration, and using straight-line program codification of the input and intermediate results, we obtain a method for the problem introduced above that is polynomial in the input length and the real degree.
Keywords and phrases: Real polynomial equation solving, polar varieties, real degree, straight-line programs, complexity

PREPRINT 96-11

Roczen, Marko

1-Semiquasihomogeneous Singularities of Hypersurfaces in Characteristic 2

Abstract:
In arbitrary characteristic different from 2, the singularities with semiquasihomogeneous equations characterized by the condition to have Saito-invariant 1 are the "classical" quasihomogeneous ones, known over the field of complex numbers as simple elliptic singularities (introduced by K. Saito). Here we find them in characteristic 2 as well: In odd dimensions and for weights E_6~ and E_7~ non-quasihomogeneous equations appear.

PREPRINT 96-12

Tammer, Klaus

Parametric Linear Complementarity Problems

Abstract:
We study linear complementarity problems depending on parameters in the right-hand side and (or) in the matrix. For the case that all elements of the right-hand side are independent parameters we give a new proof for the equivalence of three different important local properties of the corresponding solution set map in a neighbourhood of an element of its graph. For one- and multiparametric problems this equivalence does not hold and the corresponding graph may have a rather complicate structure. But we are able to show that for a generic class of linear complementarity problems depending linearly on only one real parameter the situation is much more easier.

PREPRINT 96-13

Tammer, Klaus; Tammer, Christiane; Ohlendorf, Evelin

Multicriterial Fractional Optimization

Abstract:
At first we introduce different solution concepts for general vector optimization problems and summarize some relations between them. Further, we apply these solution concepts to vectorial fractional optimization problems and show that the well-known Dinkelbach-transformation can be generalized in the sense, that even in vector optimization exact as well as approximate solutions for the original problem and for the transformed one are closely related. Finally, we discuss possibilities to handle the transformed vector optimization problem by means of parametric optimization.

PREPRINT 96-14

Siegmund-Schultze, Reinhard

Das an der Berliner Universität um 1892 ''herrschende mathematische System'' aus der Sicht des Göttingers Felix Klein:

Eine Studie über ''Raum der Wissenschaft''

aus dem Inhalt:

Kleins Kritik an der zu engen ''Schule'' und die Rolle internationaler mathematischer Kommunikation
Die Vorteile der ''kleinen Gartenstadt Göttingen'' für die Verfolgung von Kleins reformatorischen Zielen und die Rolle von Kleins amerikanischen Beziehungen
Die fehlgeschlagene Berufung Kleins nach Berlin um 1890
Anhang: Ein Brief Kleins an Friedrich Althoff vom 6. Januar 1892

PREPRINT 96-15

Lamour, René; März, Roswitha; Winkler, Renate

How Floquet-theory applies to differential-algebraic equations

Abstract:
Local stability of periodic solutions is established by means of a corresponding Floquet-theory for index-1 differential-algebraic equations. For this, linear differential-algebraic equations with periodic coefficients are considered in detail, and a natural notion of the monodromy matrix is figured out, which generalizes the well-known case of regular ordinary differential equations.

PREPRINT 96-16

Guddat, J.; Guerra, F.; Nowack, D.

On the Role of the Mangasarian-Fromovitz Constraint Qualification for Penalty-, Exact Penalty- and Lagrange Multiplier Methods

Abstract:
In this paper we consider three embeddings (one-parametric optimization problems) motivated by penalty, exact penalty and Lagrange multiplier methods. We give an answer to the question under which conditions these methods are successful with an arbitrarily chosen starting point. Using the theory of one-parametric optimization (the local structure of the set of stationary points and of the set of generalized critical points, singularities, structural stability, pathfollowing and jumps) the so-called Mangasarian-Fromovitz condition and its extension play an important role. The analysis shows us that the class of optimization problems for which we can surely find a stationary point using a pathfollowing procedure for the modified penalty and exact penalty embedding is much larger than the class where the Lagrange multiplier embedding is successful. For the first class, the objective may be a "really non-convex" function, but for the second one we are restricted to convex optimization problems. This fact was a surprise at least for the authors.

PREPRINT 96-17

Holzapfel, Rolf-Peter

Symplectic representation of a braid group on 3-sheeted covers of the Riemann sphere

Preface:
Why we called the class of two-dimensional Shimura varieties, which are not Hilbert modular, "Picard modular surfaces" ? In the mean time the name has been generally accepted, see e.g. Langlands (and others) [L-R]. On the one hand Picard worked on special Fuchsian systems of differential equations; on the other hand Shimura [Shi] introduced and investigated moduli spaces of abelian varieties with prescribed division algebra of endomorphisms, which are called (complex) "Shimura varieties" after some work of Deligne. One needs a chain of conclusions in a special case in order to connect both works. Picard found ad hoc on certain Riemann surfaces ordered sets of cycles, which we will call "Picard cycles" below. Quotients of integrals along these cycles solve (completely) a special Fuchsian system of differential equations. The basic solution consists of two multivalued complex functions of two variables. The multivalence can be described by the monodromy group of the system. By Picard-Lefschetz theory, actually described in Arnold (and others) [AVH], the monodromy group acts on the homology of an algebraic curve family respecting Picard cycles. In [H 95] (Lemma 2.27) we announced that the action on Picard cycles is transitive and, moreover, coincides with the action of an arithmetic unitary group U((2,1),O), O the ring of integers of an imaginary quadratic number field K. This is a key result. Namely, the unitary group is the modular group of the Shimura surface of (principally polarized) abelian threefolds with K-multiplication of type (2,1). It parametrizes via Jacobians the isomorphy classes of the Riemann surfaces Picard started with. The aim of this article is to give a complete proof of the mentioned key result. It joins some actual and old mathematics. As a consequence one gets a solution of the relative Schottky problem for smooth Galois coverings of P¹(C) (Riemann sphere) of degree 3 and genus 3.

PREPRINT 96-18

Bagh, Adib

Equilibrium in abstract economies without the lower semi-continuity of the constraint maps

Abstract:
We use graph convergence of set valued maps to show the existence of an equilibrium for an abstract economy without assuming the lower semi continuity of the constraint maps.

PREPRINT 96-19

Bank, B.; Giusti, M.; Heintz, J.; Mbakob, G. M.

Polar varieties, real equation solving and data-structures: The hypersurfacee case

Abstract:
In this paper we apply for the first time a new method for multivariate equation solving which was developed in for complex root determination to the real case. Our main result concerns the problem of finding at least one representative point for each connected component of a real compact and smooth hypersurface. The basic algorithm of yields a new method for symbolically solving zero-dimensional polynomial equation systems over the complex numbers. One feature of central importance of this algorithm is the use of a problem--adapted data type represented by the data structures arithmetic network and straight-line program (arithmetic circuit). The algorithm finds the complex solutions of any affine zero-dimensional equation system in non-uniform sequential time that is polynomial in the length of the input (given in straight--line program representation) and an adequately defined geometric degree of the equation system. Replacing the notion of geometric degree of the given polynomial equation system by a suitably defined real (or complex) degree of certain polar varieties associated to the input equation of the real hypersurface under consideration, we are able to find for each connected component of the hypersurface a representative point (this point will be given in a suitable encoding). The input equation is supposed to be given by a straight-line program and the (sequential time) complexity of the algorithm is polynomial in the input length and the degree of the polar varieties mentioned above.
Keywords: Real polynomial equation solving, polar variety, geometric degree, straight-line program, arithmetic network, complexity

PREPRINT 96-20

Henrion, René

Characterization of stability for cone increasing constraint mappings

Abstract:
We investigate stability (in terms of metric regularity) for the specific class of cone increasing constraint mappings. This class is of interest in problems with additional knowledge on some nondecreasing behavior of the constraints (e.g. in chance constraints, where the distribution function of some measure is automatically nondecreasing). It is demonstrated, how this extra information may lead to sharper characterizations. In the first part, rather general cone increasing constraint mappings are studied by exploiting criteria for metric regularity, as recently developed by Mordukhovich. The second part focusses on genericity investigations for global metric regularity (i.e. metric regularity at all feasible points) of nondecreasing constraints in finite dimensions. Applications to chance constraints are given.

PREPRINT 96-21

Jochmann, Frank

Existence of weak solutions of the drift diffusion model coupled with Maxwell s equations , erschienen in: Journal of Math. Anal. Appl. 204 (1996), 655-676

PREPRINT 96-22

Jochmann, Frank

Fractional order differentiability of the gradient of solutions to elliptic equations with mixed boundary conditions (siehe Anhang in P-98-23)

PREPRINT 96-23

Jochmann, Frank

Uniqueness and regularity for the two-dimensional drift-diffusion model for semiconductors coupled with Maxwell's equations , erschienen in; Journal of Differential Equations 147 (1998), 242-270

PREPRINT 96-24

Antonets, M. A

Initial Value Problem for Pseudodifferential Operators

PREPRINT 96-25

Dentcheva, D.; Gollmer, R.; Möller, A.; Römisch, W.; Schultz, R.

Solving the Unit Commitment Problem in Power Generation by Primal and Dual Methods

PREPRINT 96-26

Nowak, M. P.; Römisch, W.

Optimal Power Dispatch via Multistage Stochastic Programming

PREPRINT 96-27

Batgerel, B.; Hanke, M.;Zhanlav, T.

A nonstandard finite difference method for the solution of linear second order boundary value problems with nonsmooth coefficients

Abstract:
A new three point highly accurate finite difference method for solving linear second order differential equations is proposed. The coefficients of the scheme are constructed via differentiations of the differential equation. The accuracy and efficiency of the method is compared with other well-known methods.

PREPRINT 96-28

Dentcheva, Darinka

Differentiable Selections of Set-Valued Mappings with Application in Stochastic Programming

Abstract:
We consider set-valued mappings defined on a linear normed space with convex closed images in Rⁿ. Our aim is to construct selections which are (Hadamard) directionally differentiable using some approximation of the multifunction. The constructions suggested assume existence of a cone approximation given by a certain "derivative" of the mapping. The first one makes use of the properties of Steiner points. The notion of Steiner center is generalized for a class of unbounded sets, which include the polyhedral sets. The second construction defines a continuous selection through a given point of the graph of the multifunction and being Hadamard directionally differentiable at that point with derivatives belonging to the corresponding "derivative" of the multifunction. Both constructions lead to a directionally differentiable Castaing representation of measurable multifunctions with the required differentiability properties. The results are applied to obtain statements about the asymptotic behaviour of measurable selections of random sets via the delta-approach. Particularly, random sets of this kind build the solutions of two-stage stochastic programs.

PREPRINT 96-29

Petry, Thomas

On the stability of the Abramov transfer for differential-algebraic equations of index 1

Abstract:
The transfer of boundary conditions for ordinary differential equations developed by Abramov is a stable method for representing the solution spaces of linear boundary value problems. Instead of boundary value problems, matrix-valued initial value problems are solved. When integrating these differential equations, the inner independence of the columns of the solution matrix and, hence, of the solutions of the resulting linear system of equations, remains valid.

 K. Balla and R. März have generalized Abramov's transfer for homogenized index 1 differential-algebraic equations.

 In this article, a direct version of the Abramov transfer for inhomogeneous linear index-1 differential-algebraic equations is developed and the numerical stability of this method is proved.

PREPRINT 96-30

Sarlabous, Jorge Estrada ; Barcelo, Jorge Alejandro Pineiro

Decoding of codes on Picard curves

Abstract:
The Picard curves are genus three curves with a non trivial automorphism, which have been intensively studied due their connection with interesting number theoretic problems.
In 1989, R. Pellikaan obtained an algorithm decoding geometric codes up to [(d^*-1)/2]-errors, where d^* is the designed distance of the code. His algorithm is not completely effective, but recently some authors have given an effective answer to Pellikaan's algorithm using the particular features of special curves, such as the Klein quartic and the hyperelliptic curves.
In this paper we show that the Picard curves are suitable to obtain an effective answer to Pellikaan's algorithm.

Keywords: Error Correcting Codes, Picard curves, Jacobian Varieties.

PREPRINT 96-31

Gomez Bofill, Walter

Properties of an interior embedding for solving nonlinear optimization problems

PREPRINT 96-32

März, Roswitha

Managing the drift-off in numerical index-2 differential algebraic equations by projected defect corrections

Abstract:
When integrating index-2 differential-algebraic equations, the given constraint may be failed to be met due to the integration method itself and also due to numerical defects in the realization. This so-called drift-off gives rise to bad instabilities. In 1991 Ascher and Petzold proposed to manage the drift-off caused by symmetric implicit Runge-Kutta methods in Hessenberg systems by means of backprojections onto the constraint. In the present paper, this nice idea is generalized and analyzed in some detail for general index-2 differential-algebraic equations and, in particular, for quasilinear equations a(x,t)x' + g(x,t) = 0, as they arise in applications. Now the constraint under consideration is only implicitly given and the backprojection turns out to be rather a projected defect correction.

PREPRINT 96-33

Revalski, Julian P.

Densely defined selections of set-valued mappings and applications to the geometry of Banach spaces and optimization

PREPRINT 96-34

Jochmann, Frank

Qualitative behavior of weak solutions of the drift diffusion model for semiconductor devices coupled with Maxwell s equations

Abstract: The transient drift-diffusion model describing the charge transport in semiconductors is considered. Poisson's equation, which is usually used, is replaced by Maxwell's equations. The diffusion- and mobility-coefficients and the dielectric and magnetic susceptibilities may depend on the space-variables. Global existence and convergence to the thermal equilibrium is shown.
Key words:
Drift-diffusion-model for Semiconductors, Maxwell's equations, parabolic PDE nonlinearely coupled with hyperbolic system, global existence, asymptotic behavior.
AMS subject-class.: 35Q60,35L40,78A35

PREPRINT 96-35

Dentcheva, Darinka; Römisch, Werner

Optimal Power Generation under Uncertainty via Stochastic Programming

Abstract:
A power generation system comprising thermal and pumped-storage hydro plants is considered. Two kinds of models for the cost-optimal generation of electric power under uncertain load are introduced: (i) a dynamic model for the short-term operation and (ii) a power production planning model. In both cases, the presence of stochastic data in the optimization model leads to multi-stage and two-stage stochastic programs, respectively. Both stochastic programming problems involve a large number of mixed-integer (stochastic) decisions, but their constraints are loosely coupled across operating power units. This is used to design Lagrangian relaxation methods for both models, which lead to a decomposition into stochastic single unit subproblems. For the dynamic model a Lagrangian decomposition based algorithm is described in more detail. Special emphasis is put on a discussion of the duality gap, the efficient solution of the multi-stage single unit subproblems and on solving the dual problem by bundle methods for convex nondifferentiable optimization.
Keywords: hydro-thermal power system, uncertain load, stochastic programming, multi-stage, two-stage, mixed-integer, Lagrangian relaxation, bundle methods.

PREPRINT 96-36

Dentcheva, Darinka; Römisch, Werner

Differential Stability of Two-Stage Stochastic Programs

Abstract:
Two-stage stochastic programs with random right-hand side are considered. Optimal values and solution sets are regarded as mappings of the expected recourse functions and their perturbations, respectively. Conditions are identified implying that these mappings are directionally differentiable and semidifferentiable on appropriate functional spaces. Explicit formulas for the derivatives are derived. Special attention is paid to the role of a Lipschitz condition for solution sets as well as of a quadratic growth condition of the objective function.
Keywords: Two-stage stochastic programs, sensitivity analysis, directional derivatives, semidifferentiability, solution sets.
1991 MSC: 90C15, 90C31

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Letzte Änderung: 20.12.1996
Brigitte Richter richter@mathematik.hu-berlin.de