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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 +0200Geometric Methods to Solve Max-Ordering Location Problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/502
Location problems with Q (in general conflicting) criteria are considered. After reviewing previous results of the authors dealing with lexicographic and Pareto location the main focus of the paper is on max-ordering locations. In these location problems the worst of the single objectives is minimized. After discussing some general results (including reductions to single criterion problems and the relation to lexicographic and Pareto locations) three solution techniques are introduced and exemplified using one location problem class, each: The direct approach, the decision space approach and the objective space approach. In the resulting solution algorithms emphasis is on the representation of the underlying geometric idea without fully exploring the computational complexity issue. A further specialization of max-ordering locations is obtained by introducing lexicographic max-ordering locations, which can be found efficiently. The paper is concluded by some ideas about future research topics related to max-ordering location problems.Matthias Ehrgott; Horst W. Hamacher; Stefan Nickelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/502Mon, 03 Apr 2000 00:00:00 +0200Locating Least-Distance Lines in the Plane
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/503
In this paper we deal with locating a line in the plane. If d is a distance measure our objective is to find a straight line l which minimizes f(l) of g(l) (see the paper for the definition of these functions). We show that for all distance measures d derived from norms, one of the lines minimizing f(l) contains at least two of the existing facilities. For the center objective we always get an optimal line which is at maximum distance from at least three of the existing facilities. If all weights are equal, there is an optimal line which is parallel to one facet of the convex hull of the existing facilities.Anita Schöbelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/503Mon, 03 Apr 2000 00:00:00 +0200Saddle Points and Pareto Points in Multiple Objective Programming
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/504
In this paper relationships between Pareto points and saddle points in multiple objective programming are investigated. Convex and nonconvex problems are considered and the equivalence between Pareto points and saddle points is proved in both cases. The results are based on scalarizations of multiple objective programs and related linear and augmented Lagrangian functions. Partitions of the index sets of objectives and constranints are introduced to reduce the size of the problems. The relevance of the results in the context of decision making is also discussed.Matthias Ehrgott; Margaret M. Wiecekpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/504Mon, 03 Apr 2000 00:00:00 +0200Discrete Decision Problems, Multiple Criteria Optimization Classes and Lexicographic Max-Ordering
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/505
The topic of this paper are discrete decision problems with multiple criteria. We first define discrete multiple criteria decision problems and introduce a classification scheme for multiple criteria optimization problems. To do so we use multiople criteria optimization classes. The main result is a characterization of the class of lexicographic max-ordering problems by two very useful properties, reduction and regularity. Subsequently we discuss the assumptions under which the application of this specific MCO class is justified. Finally we provide (simple) solution methods to find optimal decisions in the case of discrete multiple criteria optimization problems.Matthias Ehrgottpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/505Mon, 03 Apr 2000 00:00:00 +0200Solving restricted line location problems via a dual interpretation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/506
In line location problems the objective is to find a straight line which minimizes the sum of distances, or the maximum distance, respectively to a given set of existing facilities in the plane. These problems have well solved. In this paper we deal with restricted line location problems, i.e. we have given a set in the plane where the line is not allowed to pass through. With the help of a geometric duality we solve such problems for the vertical distance and then extend these results to block norms and some of them even to arbitrary norms. For all norms we give a finite candidate set for the optimal line.Anita Schöbelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/506Mon, 03 Apr 2000 00:00:00 +0200Median hyperplanes in normed spaces - a survey -
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/510
In this survey we deal with the location of hyperplanes in n-dimensional normed spaces, i.e., we present all known results and a unifying approach to the so-called median hyperplane problem in Minkowski spaces. We describe how to find a hyperplane H minimizing the weighted sum f(H) of distances to a given, finite set of demand points. In robust statistics and operations research such an optimal hyperplane is called a median hyperplane.After summarizing the known results for the Euclidean and rectangular situation, 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 a halving one (in a sense defined below) with respect to the geiven point set. Also an independence of norm result for finding optimal hyperplanes with fixed slope will be given. Furthermore we discuss how these geometric criteria can be used for algorithmical approaches to median hyperplanes, with an extra discussion for the case of polyhedral norms. And finally a characterizatio of all smooth norms by a sharpened incidence criterion for median hyperplanes is mentioned.Anita Schöbel; Horst Martinipreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/510Mon, 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 +0200A flexible approach to location problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/499
In continous location problems we are given a set of existing facilities and we are looking for the location of one or several new facilities. In the classical approaches weights are assigned to existing facilities expressing the importance of the new facilities for the existing ones. In this paper, we consider a pointwise defined objective function where the weights are assigned to the existing facilities depending on the location of the new facility. This approach is shown to be a generalization of the median, center and centdian objective functions. In addition, this approach allows to formulate completely new location models. Efficient algorithms as well as structure results for this algebraic approach for location problems are presented. Extensions to the multifacility and restricted case are also considered.Stefan Nickel; Justo Puertoi; Antonio M. Rodriguez-Chiapreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/499Mon, 03 Apr 2000 00:00:00 +0200A geometric approach to global optimization
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/500
In this paper we consider the problem of optimizing a piecewise-linear objective function over a non-convex domain. In particular we do not allow the solution to lie in the interior of a prespecified region R. We discuss the geometrical properties of this problems and present algorithms based on combinatorial arguments. In addition we show how we can construct quite complicated shaped sets R while maintaining the combinatorial properties.Stefan Nickel; Anita Schöbelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/500Mon, 03 Apr 2000 00:00:00 +0200General Continuous Multicriteria Location Problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/501
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 as particular instances median, center or cent-dian objective functions. The paper characterizes the set of Pareto-solutions of all these multicriteria problems. An efficient algorithm for the planar case is developed and its complexity is established. the proposed approach is more general than the previously published approaches to multicriteria location problems and includes almost all of them as particular instances.Stefan Nickel; Justo Puerto; Antonio M. Rodriguez-Chia; Ansgar Weißlerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/501Mon, 03 Apr 2000 00:00:00 +0200Approximation Algorithms for Combinatorial Multicriteria Optimization Problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/512
The computational complexity of combinatorial multiple objective programming problems is investigated. NP-completeness and #P-completeness results are presented. Using two definitions of approximability, general results are presented, which outline limits for approximation algorithms. The performance of the well known tree and Christofides' heuristics for the TSP is investigated in the multicriteria case with respect to the two definitions of approximability.Matthias Ehrgottpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/512Mon, 03 Apr 2000 00:00:00 +0200Convex Operators in Vector Optimization: Directional Derivatives and the Cone of Decrease Directions
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/514
The paper is devoted to the investigation of directional derivatives and the cone of decrease directions for convex operators on Banach spaces. We prove a condition for the existence of directional derivatives which does not assume regularity of the ordering cone K. This result is then used to prove that for continuous convex operators the cone of decrease directions can be represented in terms of the directional derivatices . Decrease directions are those for which the directional derivative lies in the negative interior of the ordering cone K. Finally, we show that the continuity of the convex operator can be replaced by its K-boundedness.Alexander L. Topchishvili; Vilhelm G. Maisuradze; Matthias Ehrgottpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/514Mon, 03 Apr 2000 00:00:00 +0200