## Report in Wirtschaftsmathematik (WIMA Report)

- 140
- Variants of the Shortest Path Problem (2011)
- The shortest path problem in which the \((s,t)\)-paths \(P\) of a given digraph \(G =(V,E)\) are compared with respect to the sum of their edge costs is one of the best known problems in combinatorial optimization. The paper is concerned with a number of variations of this problem having different objective functions like bottleneck, balanced, minimum deviation, algebraic sum, \(k\)-sum and \(k\)-max objectives, \((k_1, k_2)-max, (k_1, k_2)\)-balanced and several types of trimmed-mean objectives. We give a survey on existing algorithms and propose a general model for those problems not yet treated in literature. The latter is based on the solution of resource constrained shortest path problems with equality constraints which can be solved in pseudo-polynomial time if the given graph is acyclic and the number of resources is fixed. In our setting, however, these problems can be solved in strongly polynomial time. Combining this with known results on \(k\)-sum and \(k\)-max optimization for general combinatorial problems, we obtain strongly polynomial algorithms for a variety of path problems on acyclic and general digraphs.

- 139a
- Asymptotic Order of the Parallel Volume Difference (2012)
- In this paper we investigate the asymptotic behaviour of the parallel volume of fixed non-convex bodies in Minkowski spaces as the distance \(r\) tends to infinity. We will show that the difference of the parallel volume of the convex hull of a body and the parallel volume of the body itself can at most have order \(r^{d-2}\) in a \(d\)-dimensional space. Then we will show that in Euclidean spaces this difference can at most have order \(r^{d-3}\). These results have several applications, e.g. we will use them to compute the derivative of \(f_\mu(rK)\) in \(r = 0\), where \(f_\mu\) is the Wills functional or a similar functional, \(K\) is a body and \(rK\) is the Minkowski-product of \(r\) and \(K\). Finally we present applications concerning Brownian paths and Boolean models and derive new proofs for formulae for intrinsic volumes.

- 139
- Asymptotic order of the parallel volume difference (2011)
- In this paper we continue the investigation of the asymptotic behavior of the parallel volume in Minkowski spaces as the distance tends to infinity that was started in [13]. We will show that the difference of the parallel volume of the convex hull of a body and the parallel volume of the body itself can at most have order \(r^{d-2}\) in dimension \(d\). Then we will show that in the Euclidean case this difference can at most have order \(r^{d-3}\). We will also examine the asymptotic behavior of the derivative of this difference as the distance tends to infinity. After this we will compute the derivative of \(f_\mu (rK)\) in \(r\), where \(f_\mu\) is the Wills functional or a similar functional, \(K\) is a fixed body and \(rK\) is the Minkowski-product of \(r\) and \(K\). Finally we will use these results to examine Brownian paths and Boolean models and derive new proofs for formulae for intrinsic volumes.

- 138
- A uniform central limit theorem for neural network based autoregressive processes with applications to change-point analysis (2011)
- We consider an autoregressive process with a nonlinear regression function that is modeled by a feedforward neural network. We derive a uniform central limit theorem which is useful in the context of change-point analysis. We propose a test for a change in the autoregression function which - by the uniform central limit theorem - has asymptotic power one for a large class of alternatives including local alternatives.

- 137
- Testing for parameter stability in nonlinear autoregressive models (2011)
- In this paper we develop testing procedures for the detection of structural changes in nonlinear autoregressive processes. For the detection procedure we model the regression function by a single layer feedforward neural network. We show that CUSUM-type tests based on cumulative sums of estimated residuals, that have been intensively studied for linear regression, can be extended to this case. The limit distribution under the null hypothesis is obtained, which is needed to construct asymptotic tests. For a large class of alternatives it is shown that the tests have asymptotic power one. In this case, we obtain a consistent change-point estimator which is related to the test statistics. Power and size are further investigated in a small simulation study with a particular emphasis on situations where the model is misspecified, i.e. the data is not generated by a neural network but some other regression function. As illustration, an application on the Nile data set as well as S&P log-returns is given.

- 133
- On a Cardinality Constrained Multicriteria Knapsack Problem (2011)
- We consider a variant of a knapsack problem with a fixed cardinality constraint. There are three objective functions to be optimized: one real-valued and two integer-valued objectives. We show that this problem can be solved efficiently by a local search. The algorithm utilizes connectedness of a subset of feasible solutions and has optimal run-time.

- 131
- Generalized Multiple Objective Bottleneck Problems (2010)
- We consider multiple objective combinatiorial optimization problems in which the first objective is of arbitrary type and the remaining objectives are either bottleneck or k-max objective functions. While the objective value of a bottleneck objective is determined by the largest cost value of any element in a feasible solution, the kth-largest element defines the objective value of the k-max objective. An efficient solution approach for the generation of the complete nondominated set is developed which is independent of the specific combinatiorial problem at hand. This implies a polynomial time algorithm for several important problem classes like shortest paths, spanning tree, and assignment problems with bottleneck objectives which are known to be NP-hard in the general multiple objective case.

- 128
- Universal Shortest Paths (2010)
- We introduce the universal shortest path problem (Univ-SPP) which generalizes both - classical and new - shortest path problems. Starting with the definition of the even more general universal combinatorial optimization problem (Univ-COP), we show that a variety of objective functions for general combinatorial problems can be modeled if all feasible solutions have the same cardinality. Since this assumption is, in general, not satisfied when considering shortest paths, we give two alternative definitions for Univ-SPP, one based on a sequence of cardinality contrained subproblems, the other using an auxiliary construction to establish uniform length for all paths between source and sink. Both alternatives are shown to be (strongly) NP-hard and they can be formulated as quadratic integer or mixed integer linear programs. On graphs with specific assumptions on edge costs and path lengths, the second version of Univ-SPP can be solved as classical sum shortest path problem.

- 126
- Weak Dependence of Functional INGARCH Processes (2010)
- We introduce a class of models for time series of counts which include INGARCH-type models as well as log linear models for conditionally Poisson distributed data. For those processes, we formulate simple conditions for stationarity and weak dependence with a geometric rate. The coupling argument used in the proof serves as a role model for a similar treatment of integer-valued time series models based on other types of thinning operations.

- 124
- Maximum Likelihood Estimators for Markov Switching Autoregressive Processes with ARCH Component (2009)
- We consider a mixture of AR-ARCH models where the switching between the basic states of the observed time series is controlled by a hidden Markov chain. Under simple conditions, we prove consistency and asymptotic normality of the maximum likelihood parameter estimates combining general results on asymptotics of Douc et al (2004) and of geometric ergodicity of Franke et al (2007).