Report in Wirtschaftsmathematik (WIMA Report)
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1
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.
5
Given a finite set of points in the plane and a forbidden region R, we want to find a point X not an element of int(R), such that the weighted sum to all given points is minimized. This location problem is a variant of the well-known Weber Problem, where we measure the distance by polyhedral gauges and allow each of the weights to be positive or negative. The unit ball of a polyhedral gauge may be any convex polyhedron containing the origin. This large class of distance functions allows very general (practical) settings - such as asymmetry - to be modeled. Each given point is allowed to have its own gauge and the forbidden region R enables us to include negative information in the model. Additionally the use of negative and positive weights allows to include the level of attraction or dislikeness of a new facility. Polynomial algorithms and structural properties for this global optimization problem (d.c. objective function and a non-convex feasible set) based on combinatorial and geometrical methods are presented.
54
Value Preserving Strategies and a General Framework for Local Approaches to Optimal Portfolios
(1999)
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.
55
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.
32
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.
30
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.
50
In this paper a new trend is introduced into the field of multicriteria location problems. We combine the robustness approach using the minmax regret criterion together with Pareto-optimality. We consider the multicriteria Weber location problem which consists of simultaneously minimizing a number of weighted sum-distance functions and the set of Pareto-optimal locations as its solution concept. For this problem, we characterize the Pareto-optimal solutions within the set of robust locations for the original weighted sum-distance functions. These locations have both the properties of stability and non-domination which are required in robust and multicriteria programming.
20
Ramsey Numbers of K_m versus (n,k)-graphs and the Local Density of Graphs not Containing a K_m
(1999)
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.
58
13
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.