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Wed, 04 Oct 2000 00:00:00 +0200Wed, 04 Oct 2000 00:00:00 +0200Some Applications of Impulse Control in Mathematical Finance
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1143
We consider three applications of impulse control in financial mathematics, a cash management problem, optimal control of an exchange rate, and portfolio optimisation under transaction costs. We sketch the different ways of solving these problems with the help of quasi-variational inequalities. Further, some viscosity solution results are presented.Ralf Kornpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1143Wed, 04 Oct 2000 00:00:00 +0200Hyperplane transversals of homothetical, centrally symmetric polytopes
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1136
Let P c R^n, n >= 2, be a centrally symmetric, convex n-polytope with 2r vertices, and P be a family of m >= n + 1 homothetical copies of P. We show that a hyperplane transversal of all members of P (it it exists) can be found in O(rm) time.Horst Martini; Anita Schöbelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1136Wed, 30 Aug 2000 00:00:00 +0200A Fuzzy Programming Approach to Multicriteria Facility Location Problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1124
Facility Location Problems are concerned with the optimal location of one or several new facilities, with respect to a set of existing ones. The objectives involve the distance between new and existing facilities, usually a weighted sum or weighted maximum. Since the various stakeholders (decision makers) will have different opinions of the importance of the existing facilities, a multicriteria problem with several sets of weights, and thus several objectives, arises. In our approach, we assume the decision makers to make only fuzzy comparisons of the different existing facilities. A geometric mean method is used to obtain the fuzzy weights for each facility and each decision maker. The resulting multicriteria facility location problem is solved using fuzzy techniques again. We prove that the final compromise solution is weakly Pareto optimal and Pareto optimal, if it is unique, or under certain assumptions on the estimates of the Nadir point. A numerical example is considered to illustrate the methodology.Matthias Ehrgott; Rakesh Vermapreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1124Tue, 29 Aug 2000 00:00:00 +0200Multicriteria Ordered Weber Problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1126
In this paper we deal with the determination of the whole set of Pareto-solutions of location problems with respect to Q general criteria.These criteria include median, center or cent-dian objective functions as particular instances.The paper characterizes the set of Pareto-solutions of a these multicriteria problems. An efficient algorithm for the planar case is developed and its complexity is established. Extensions to higher dimensions as well as to the non-convexcase are also considered.The proposed approach is more general than the previously published approaches to multi-criteria location problems and includes almost all of them as particular instances.Stefan Nickel; Justo Puerto; Antonio M. Rodriguez-Chiapreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1126Tue, 29 Aug 2000 00:00:00 +0200On value preserving and growth optimal portfolios
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1133
In a discrete-time financial market setting, the paper relates various concepts introduced for dynamic portfolios (both in discrete and in continuous time). These concepts are: value preserving portfolios, numeraire portfolios, interest oriented portfolios, and growth optimal portfolios. It will turn out that these concepts are all associated with a unique martingale measure which agrees with the minimal martingale measure only for complete markets.Ralf Korn; Manfred Schälpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1133Tue, 29 Aug 2000 00:00:00 +0200Value Preserving Strategies and a General Framework for Local Approaches to Optimal Portfolios
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1120
We present some new general results on the existence and form of value preserving portfolio strategies in a general semimartingale setting. The concept of value preservation will be derived via a mean-variance argument. It will also be embedded into a framework for local approaches to the problem of portfolio optimisation.Ralf Kornpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1120Mon, 28 Aug 2000 00:00:00 +0200Portfolio management and market risk quantification using neural networks
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1121
We discuss how neural networks may be used to estimate conditional means, variances and quantiles of nancial time series nonparametrically. These estimates may be used to forecast, to derive trading rules and to measure market risk.Jürgen Frankepreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1121Mon, 28 Aug 2000 00:00:00 +0200Min-Max Formulation of the Balance Number in Multiobjetive Global Optimization
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/515
The notion of the balance number introduced in [3,page 139] through a certain set contraction procedure for nonscalarized multiobjective global optimization is represented via a min-max operation on the data of the problem. This representation yields a different computational procedure for the calculation of the balance number and allows us to generalize the approach for problems with countably many performance criteria.Matthias Ehrgott; Efim A. Galperinpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/515Mon, 03 Apr 2000 00:00:00 +0200Zur Ermittlung des Verkehrswerts bebauter Grundstücke in Kaiserslautern
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/482
Anhand des vom Gutachterausschuß der Stadt Kaiserlautern zur Verfügung gestellten Datenmaterials soll untersucht werden, welche Faktoren den Verkehrswert eines bebauten Grundstücks beeinflussen. Mit diesen Erkenntnissen soll eine möglichst einfache Formel ermittelt werden, die eine Schätzung für den Verkehrswert liefert, und die dabei die in der Vergangenheit erzielten Kaufpreise berücksichtigt. Für die Lösung dieser Aufgabe bietet sich das Verfahren der multiplen linearen Regression an. Auf die theoretischen Grundlagen soll hier nicht näher eingegangen werden, man findet sie in jedem Buch über mathematische Statistik, oder in [1]. Bei der Analyse der Daten wurde im großen und ganzen der Weg eingeschlagen, den Angelika Schwarz in [1] beschreibt. Ihre Ergebnisse lassen sich jedoch nicht direkt übertragen, da die dort betrachteten Grundstücke unbebaut waren. Da bei der statistischen Auswertung großer Datenmengen ein immenser Rechenaufwand anfällt, ist es unverzichtbar, professionelle statistische Software einzusetzen. Es stand das Programm S-Plus 2.0 (PC-Version für Windows) zur Verfügung. Sämtliche Berechnungen und alle Grafiken in diesem Bericht wurden in S-Plus erstellt.Axel Krebspreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/482Mon, 03 Apr 2000 00:00:00 +0200A reduction algorithm for integer multiple objective linear programs
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/483
We consider a multiple objective linear program (MOLP) max{Cx|Ax = b,x in N_{0}^{n}} where C = (c_ij) is the p x n - matrix of p different objective functions z_i(x) = c_{i1}x_1 + ... + c_{in}x_n , i = 1,...,p and A is the m x n - matrix of a system of m linear equations a_{k1}x_1 + ... + a_{kn}x_n = b_k , k=1,...,m which form the set of constraints of the problem. All coefficients are assumed to be natural numbers or zero. The set M of admissable solutions {hat x} is an admissible solution such that there exists no other admissable solution x' with C{hat x} Cx'. The efficient solutions play the role of optimal solutions for the MOLP and it is our aim to determine the set of all efficient solutionsDietmar Schweigert; Peter Neumayerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/483Mon, 03 Apr 2000 00:00:00 +0200A Characterization of Lexicographic Max-Ordering Solutions
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/484
In this paper we give the definition of a solution concept in multicriteria combinatorial optimization. We show how Pareto, max-ordering and lexicographically optimal solutions can be incorporated in this framework. Furthermore we state some properties of lexicographic max-ordering solutions, which combine features of these three kinds of optimal solutions. Two of these properties, which are desirable from a decision maker" s point of view, are satisfied if and only of the solution concept is that of lexicographic max-ordering.Matthias Ehrgottpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/484Mon, 03 Apr 2000 00:00:00 +0200Planar Location Problems with Line Barriers
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/485
The Weber Problem for a given finite set of existing facilities {cal E}x = {Ex_1,Ex_2, ... ,Ex_M} subset R^2 with positive weights w_m (m = 1, ... ,M) is to find a new fcility X* such that sum_{m=1}^{M} w_{m}d(X,Ex_m) is minimized for some distance function d. A variation of this problem is obtained of the existing facilities are situated on two sides of a linear barrier. Such barriers like rivers, highways, borders or mountain ranges are frequently encountered in practice. Structural results as well as algorithms for this non-convex optimization problem depending on the distance function and on the number and location of passages through the barrier are presented. A reduction to convex optimization problems is used to derive efficient algorithms.Kathrin Klamrothpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/485Mon, 03 Apr 2000 00:00:00 +0200A unified approach to network location problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/486
In this paper we introduce a new type of single facility location problems on networks which includes as special cases most of the classical criteria in the literature. Structural results as well as a finite dominationg set for the optimal locations are developed. Also the extension to the multi-facility case is discussed.Stefan Nickel; Justo Puertopreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/486Mon, 03 Apr 2000 00:00:00 +0200Multicriteria network location problems with sumb objectives
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/487
In this paper network location problems with several objectives are discussed, where every single objective is a classical median objective function. We will lock at the problem of finding Pareto optimal locations and lexicographically optimal locations. It is shown that for Pareto optimal locations in undirected networks no node dominance result can be shown. Structural results as well as efficient algorithms for these multi-criteria problems are developed. In the special case of a tree network a generalization of Goldman's dominance algorithm for finding Pareto locations is presented.Horst W. Hamacher; Stefan Nickel; Martine Labbepreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/487Mon, 03 Apr 2000 00:00:00 +0200Multiple objective programming with piecewise linear functions
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/488
An approach to generating all efficient solutions of multiple objective programs with piecewise linear objective functions and linear constraints is presented. The approach is based on the decomposition of the feasible set into subsets, referred to as cells, so that the original problem reduces to a series of lenear multiple objective programs over the cells. The concepts of cell-efficiency and complex-efficiency are introduced and their relationship with efficiency is examined. A generic algorithm for finding efficent solutions is proposed. Applications in location theory as well as in worst case analysis are highlighted.Stefan Nickel; M. Wiecekpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/488Mon, 03 Apr 2000 00:00:00 +0200Error bounds for the approximative solution of restricted planar location problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/489
Facility location problems in the plane play an important role in mathematical programming. When looking for new locations in modeling real-word problems, we are often confronted with forbidden regions, that are not feasible for the placement of new locations. Furthermore these forbidden regions may habe complicated shapes. It may be more useful or even necessary to use approcimations of such forbidden regions when trying to solve location problems. In this paper we develop error bounds for the approximative solution of restricted planar location problems using the so called sandwich algorithm. The number of approximation steps required to achieve a specified error bound is analyzed. As examples of these approximation schemes, we discuss round norms and polyhedral norms. Also computational tests are included.Stefan Nickel; Barbara Käferpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/489Mon, 03 Apr 2000 00:00:00 +0200Median hyperplanes in normed spaces
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/490
In this paper we deal with the location of hyperplanes in n-dimensional normed spaces. If d is a distance measure, our objective is to find a hyperplane H which minimizes f(H) = sum_{m=1}^{M} w_{m}d(x_m,H), where w_m ge 0 are non-negative weights, x_m in R^n, m=1, ... ,M demand points and d(x_m,H)=min_{z in H} d(x_m,z) is the distance from x_m to the hyperplane H. In robust statistics and operations research such an optimal hyperplane is called a median hyperplane. We show that for all distance measures d derived from norms, one of the hyperplanes minimizing f(H) is the affine hull of n of the demand points and, moreover, that each median hyperplane is (ina certain sense) a halving one with respect to the given point set.Anita Schöbel; H. Martinipreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/490Mon, 03 Apr 2000 00:00:00 +0200Classification of Location Problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/491
There are several good reasons to introduce classification schemes for optimization problems including, for instance, the ability for concise problem statement opposed to verbal, often ambiguous, descriptions or simple data encoding and information retrieval in bibliographical information systems or software libraries. In some branches like scheduling and queuing theory classification is therefore a widely accepted and appreciated tool. The aim of this paper is to propose a 5-position classification which can be used to cover all location problems. We will provide a list of currentliy available symbols and indicate its usefulness in a - necessarily non-comprehensive - list of classical location problems. The classification scheme is in use since 1992 and has since proved to be useful in research, software development, classroom, and for overview articles.Horst W. Hamacher; Stefan Nickel; Anja Schneiderpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/491Mon, 03 Apr 2000 00:00:00 +0200On the number of Criteria Needed to Decide Pareto Optimality
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/492
In this paper we prove a reduction result for the number of criteria in convex multiobjective optimization. This result states that to decide wheter a point x in the decision space is pareto optimal it suffices to consider at most n? criteria at a time, where n is the dimension of the decision space. The main theorem is based on a geometric characterization of pareto, strict pareto and weak pareto solutionsMatthias Ehrgott; Stefan Nickelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/492Mon, 03 Apr 2000 00:00:00 +0200Ramsey Numbers of K_m versus (n,k)-graphs and the Local Density of Graphs not Containing a K_m
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/493
In this paper generalized Ramsey numbers of complete graphs K_m versus the set langle ,n,k angle of (n,k)-graphs are investigated. The value of r(K_m,langle n,k angle) is given in general for (relative to n) values of k small compared to n using a correlation with Turan numbers. These generalized Ramsey numbers con be used to determine the local densities of graphs not containing a subgraph K_m.Kathrin Klamroth; Ingrid Mengersenpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/493Mon, 03 Apr 2000 00:00:00 +0200Planar Location Problems with Barriers under Polyhedral Gauges
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/494
The Weber problem for a given finite set of existing facilities {cal E}x = {Ex_1,Ex_2, ... ,Ex_M} subset R^2 with positive weights w_m (m = 1, ... ,M) is to find a new facility X* in R^2 such that sum_{m=1}^{M} w_{m}d(X,Ex_m) is minimized for some distance function d. In this paper we consider distances defined by polyhedral gauges. A variation of this problem is obtained if barriers are introduced which are convex polygonal subsets of the plane where neither location of new facilities nor traveling is allowed. Such barriers like lakes, military regions, national parks or mountains are frequently encountered in practice.From a mathematical point of view barrier problems are difficult, since the prensence of barriers destroys the convexity of the objective function. Nevertheless, this paper establishes a descretization result: One of the grid points in the grid defined by the existing facilities and the fuundamental directions of the gauge distances can be proved to be an optimal location. Thus the barrier problem can be solved with a polynomial algorithm.Horst W. Hamacher; Kathrin Klamrothpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/494Mon, 03 Apr 2000 00:00:00 +0200Bootstrap of kernel smoothing in nonlinear time series
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/495
Kernel smoothing in nonparametric autoregressive schemes offers a powerful tool in modelling time series. In this paper it is shown that the bootstrap can be used for estimating the distribution of kernel smoothers. This can be done by mimicking the stochastic nature of the whole process in the bootstrap resampling or by generating a simple regression model. Consistency of these bootstrap procedures will be shown.Jürgen Franke; Kreiss J.-P.; E. Mammenpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/495Mon, 03 Apr 2000 00:00:00 +0200An Interior Point Method for Multifacility Location Problems with Forbidden Regions
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/496
In this paper we consider generalizations of multifacility location problems in which as an additional constraint the new facilities are not allowed to be located in a presprcified region. We propose several different solution schemes for this non-convex optimization problem. These include a linear programming type approach, penalty approaches and barrier approaches. Moreover, structural results as well as illustratrive examples showing the difficulties of this problem are presentedStefan Nickel; Jörg Fliegepreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/496Mon, 03 Apr 2000 00:00:00 +0200Minimal paths on ordered graphs
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/497
To present the decision maker's (DM) preferences in multicriteria decision problems as a partially ordered set is an effective method to catch the DM's purpose and avoid misleading results. Since our paper is focused on minimal path problems, we regard the ordered set of edges (E,=). Minimal paths are defined in repect to power-ordered sets which provides an essential tool to solve such problems. An algorithm to detect minimal paths on a multicriteria minimal path problem is presentedUlrike Bossong; Dietmar Schweigertpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/497Mon, 03 Apr 2000 00:00:00 +0200On Burdick's symmetry problem
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/498
Let P be a probability measure of the real line R such that each of the product measures P^{otimes n} assigns the value 1/2 to every half space in R^{n} having the origin as a boundary point. Then P is symmetric.Example: A strictly stable law on R is symmetric iff it has median zero. The treated symmetry problem is related to the problem of characterizing the distribution of X_1 by the distribution of (X_2 + X_1, ... ,X_n + X_1), with X_1, ... ,X_n being independent and identically distributed random variables.Lutz Mattnerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/498Mon, 03 Apr 2000 00:00:00 +0200