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We present a formal procedure for structure-preserving model reduction of linear second-order control problems that appear in a variety of physical contexts, e.g., vibromechanical systems or electrical circuit design. Typical balanced truncation methods that project onto the subspace of the largest Hankel singular values fail to preserve the problem's physical structure and may suffer from lack of stability. In this paper we adopt the framework of generalized Hamiltonian systems that covers the class of relevant problems and that allows for a generalization of balanced truncation to second-order problems.
It turns out that the Hamiltonian structure, stability and passivity are preserved if the truncation is done by imposing a holonomic constraint on the system rather than standard Galerkin projection.
The Steiner connectivity problem is a generalization of
the Steiner tree problem. It consists in finding a minimum cost set of
simple paths to connect a subset of nodes in an undirected graph.
We show that important polyhedral and algorithmic results on the
Steiner tree problem carry over to the Steiner connectivity problem,
namely, the Steiner cut and the Steiner partition inequalities, as
well as the associated polynomial time separation algorithms, can be
generalized. Similar to the Steiner tree case, a certain directed
formulation, which is stronger than the natural undirected one,
plays a central role.
For many fundamental cooperative cost sharing games, especially when costs are supermodular, it is known that Moulin mechanisms inevitably suffer from poor budget balance factors. Mehta, Roughgarden, and Sundararajan recently introduced acyclic mechanisms, which achieve a slightly weaker notion of group-strategyproofness, but leave more flexibility to improve upon the approximation guarantees with respect to budget balance and social cost.
In this paper, we provide a very simple but powerful method for turning any rho-approximation algorithm for a combinatorial optimization problem into a rho-budget balanced acyclic mechanism. Hence, we show that there is no gap between the best possible approximation guarantees of full-knowledge approximation algorithms and weakly group-strategyproof cost sharing mechanisms.
The applicability of our method is demonstrated by deriving mechanisms for scheduling and network design problems which beat the best possible budget balance factors of Moulin mechanisms. By elaborating our framework, we provide means to construct weakly group-strategyproof mechanisms with approximate social cost. The mechanisms we develop for completion time scheduling problems perform surprisingly well by achieving the first constant budget balance and social cost factors.
When managing energy or weather related risk often only imperfect hedging instruments are available. In the first part we illustrate problems arising with imperfect hedging by studying a toy model. We consider an airline’s problem with covering income risk due to fluctuating kerosene prices by investing into futures written on heating oil with closely correlated price dynamics. In the second part we outline recent results on exponential utility based cross hedging concepts. They highlight in a generalization of the Black-Scholes delta hedge formula to incomplete markets. Its derivation is based on a purely stochastic approach of utility maximization. It interprets stochastic control problems in the BSDE language, and profits from the power of the stochastic calculus of variations.
We consider backward stochastic differential equations with drivers of quadratic growth (qgBSDE). We prove several statements concerning path regularity and stochastic smoothness of the solution processes of the qgBSDE, in particular we prove an extension of Zhang's path regularity theorem to the quadratic growth setting. We give explicit convergence rates for the difference between the solution of a qgBSDE and its truncation, filling an important gap in numerics for qgBSDE. We give an alternative proof of second order Malliavin differentiability for BSDE with drivers that are Lipschitz continuous (and differentiable), and then derive the same result for qgBSDE.
We show that the spectrum of linear delay differential equations with
large delay splits into two different parts. One part, called the
strong spectrum, converges to isolated points when the delay parameter
tends to infinity. The other part, called the pseudocontinuous spectrum,
accumulates near criticality and converges after rescaling to a set
of spectral curves, called the asymptotic continuous spectrum. We
show that the spectral curves and strong spectral points provide a
complete description of the spectrum for sufficiently large delay
and can be comparatively easily calculated by approximating expressions.
Local existence, uniqueness and smooth dependence for nonsmooth quasilinear parabolic problems
(2009)
We prove local existence, uniqueness, Hölder regularity in space and time, and smooth dependence in Hölder spaces for a general class of quasilinear parabolic initial boundary value problems with nonsmooth data.
As a result the gap between low smoothness of the data, which is typical for
many applications, and high smoothness of the solutions, which is necessary
for the applicability of differential calculus to abstract formulations of the
initial boundary value problems, has been closed. The theory works for any
space dimension, and the nonlinearities are allowed to be nonlocal and to
have any growth. The main tools are new maximal regularity results [19, 20]
in Sobolev–Morrey spaces for linear parabolic initial boundary value problems
with nonsmooth data, linearization techniques and the Implicit Function Theorem.
In this paper we study a certain cardinality constrained packing integer program which is motivated by the problem of dimensioning a cut in a two-layer network. We prove NP-hardness and consider the facial structure of the corresponding polytope. We provide a complete description for the smallest nontrivial case and develop two general classes of facet-defining inequalities. This approach extends the
notion of the well known cutset inequalities to two network layers.
In this paper, we present a model-based optimization approach for the design of multi-layer networks. The proposed framework is based on a series of increasingly abstract models – from a general technical system model to a problem specific mathematical model – which are used in a planning cycle to optimize the multi-layer networks. In a case study we show how central design questions for an IP-over-WDM network architecture can be answered
using this approach. Based on reference networks from the German research project EIBONE, we investigate the influence of various planning parameters on the total design cost. This includes a comparison of point-to-point vs. transparent optical layer architectures, different traffic distributions, and the use of PoS vs. Ethernet interfaces.
We study superhedging of contingent claims with physical delivery in a discrete-time market model with convex transaction costs. Our model extends Kabanov's currency market model by allowing for nonlinear illiquidity effects. We show that an appropriate generalization of Schachermayer's robust no arbitrage condition implies that the set of claims hedgeable with zero cost is closed in probability. Combined with classical techniques of convex analysis, the closedness yields a dual characterization of premium processes that are sufficient to superhedge a given claim process. We also extend the fundamental theorem of asset pricing for general conical models.