Preprints: Veröffentlichungsliste 1996
Sample path large deviations for super-Brownian motion
Henrion, René; Römisch, Werner
Metric regularity and quantitative stability in stochastic programs
with probabilistic constraints
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.
A global L*-gradient estimate on weak solutions to nonlinear stationary
Navier- Stokes equations under mixed boundary conditions
März, Roswitha; Tischendorf, Caren
Recent results in solving index 2 differential-algebraic equations in
to appear in SIAM J. Sci. Stat. Comput., 18 (1), 1997.
Keywords: Differential-algebraic equations, index 2, circuit simulation,
IVP, numerical integration, BDF, defect correction.
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
Gomez Bofill, Walter
Vector optimization: Singularities, regularizations
We discuss three scalarizations of the multiobjective optimization from
the point of view of the parametric optimization. We analize three important
What kind of singularities may appear in the different parametrizations
Regularizations in the sense of Jongen, Jonker and Twilt, and in the sense
of Kojima and Hirabayashi.
The Mangasarian- Fromovitz Constraint Qualification for the first parametrization.
Dentcheva, D.; Möller, A.; Reeh, P.; Römisch, W.; Schultz,
R.; Schwarzbach, G.; Thomas, J.
Optimale Blockauswahl bei der Kraftwerkseinsatzplanung
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
Fandom, Noubiap R.
On Modifications of the Standard Embedding in Nonlinear Optimization
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.
Geometric aspects of Fleming-Viot and superprocesses
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.
Moderate deviations and functional LIL for super Brownian motion
Bank, B.; Giusti, M.; Heintz, J.; Mandel, R.; Mbakop, G. M.
Polar Varieties and Efficient Real Equation Solving: The Hypersurface
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
Replacing the affine degree of the equation system by a suitably defined
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
1-Semiquasihomogeneous Singularities of Hypersurfaces in Characteristic
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.
Parametric Linear Complementarity Problems
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.
Tammer, Klaus; Tammer, Christiane; Ohlendorf, Evelin
Multicriterial Fractional Optimization
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.
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
Die Vorteile der ''kleinen Gartenstadt Göttingen'' für die
Verfolgung von Kleins reformatorischen Zielen und die Rolle von Kleins
Die fehlgeschlagene Berufung Kleins nach Berlin um 1890
Anhang: Ein Brief Kleins an Friedrich Althoff vom 6. Januar 1892
Lamour, René; März, Roswitha; Winkler, Renate
How Floquet-theory applies to differential-algebraic equations
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
Guddat, J.; Guerra, F.; Nowack, D.
On the Role of the Mangasarian-Fromovitz Constraint Qualification for
Penalty-, Exact Penalty- and Lagrange Multiplier Methods
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
Symplectic representation of a braid group on 3-sheeted covers of the
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.
Equilibrium in abstract economies without the lower semi-continuity
of the constraint maps
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.
Bank, B.; Giusti, M.; Heintz, J.; Mbakob, G. M.
Polar varieties, real equation solving and data-structures: The hypersurfacee
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
Characterization of stability for cone increasing constraint mappings
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
Existence of weak solutions of the drift diffusion model coupled with
Maxwell s equations , erschienen in: Journal of Math. Anal. Appl. 204
Fractional order differentiability of the gradient of solutions to elliptic
equations with mixed boundary conditions (siehe Anhang in P-98-23)
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
Antonets, M. A
Initial Value Problem for Pseudodifferential Operators
Dentcheva, D.; Gollmer, R.; Möller, A.; Römisch, W.; Schultz,
Solving the Unit Commitment Problem in Power Generation by Primal and
Nowak, M. P.; Römisch, W.
Optimal Power Dispatch via Multistage Stochastic Programming
Batgerel, B.; Hanke, M.;Zhanlav, T.
A nonstandard finite difference method for the solution of linear second
order boundary value problems with nonsmooth coefficients
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
Differentiable Selections of Set-Valued Mappings with Application in
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.
On the stability of the Abramov transfer for differential-algebraic
equations of index 1
K. Balla and R. März have generalized Abramov's transfer for
homogenized index 1 differential-algebraic equations.
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,
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.
Sarlabous, Jorge Estrada ; Barcelo, Jorge Alejandro Pineiro
Decoding of codes on Picard curves
Keywords: Error Correcting Codes, Picard curves, Jacobian Varieties.
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
In this paper we show that the Picard curves are suitable to obtain an
effective answer to Pellikaan's algorithm.
Gomez Bofill, Walter
Properties of an interior embedding for solving nonlinear optimization
Managing the drift-off in numerical index-2 differential algebraic equations
by projected defect corrections
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.
Revalski, Julian P.
Densely defined selections of set-valued mappings and applications to
the geometry of Banach spaces and optimization
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.
Drift-diffusion-model for Semiconductors, Maxwell's equations, parabolic
PDE nonlinearely coupled with hyperbolic system, global existence, asymptotic
AMS subject-class.: 35Q60,35L40,78A35
Dentcheva, Darinka; Römisch, Werner
Optimal Power Generation under Uncertainty via Stochastic Programming
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
Keywords: hydro-thermal power system, uncertain load, stochastic programming,
multi-stage, two-stage, mixed-integer, Lagrangian relaxation, bundle methods.
Dentcheva, Darinka; Römisch, Werner
Differential Stability of Two-Stage Stochastic Programs
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
Letzte Änderung: 20.12.1996
Brigitte Richter email@example.com