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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.

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.

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.

Discrete Decision Problems, Multiple Criteria Optimization Classes and Lexicographic Max-Ordering
(1999)

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.

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.

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.

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 solutions

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.

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.

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.

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 presented

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 presented

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.

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.

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.

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.

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.

Convex Operators in Vector Optimization: Directional Derivatives and the Cone of Decrease Directions
(1999)

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.

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.

In this paper we consider the problem of locating one new facility in the plane with respect to a given set of existing facility where a set of polygonal barriers restricts traveling. This non-convex optimization problem can be reduced to a finite set of convex subproblems if the objective function is a convex function of the travel distances between the new and the existing facilities (like e.g. the Median and Center objective functions). An exact Algorithm and a heuristic solution procedure based on this reduction result are developed.

Let rC and rD be two convexdistance funtions in the plane with convex unit balls C and D. Given two points, p and q, we investigate the bisector, B(p,q), of p and q, where distance from p is measured by rC and distance from q by rD. We provide the following results. B(p,q) may consist of many connected components whose precise number can be derived from the intersection of the unit balls, C nd D. The bisector can contain bounded or unbounded 2-dimensional areas. Even more surprising, pieces of the bisector may appear inside the region of all points closer to p than to q. If C and D are convex polygons over m and m vertices, respectively, the bisector B(p,q) can consist of at most min(m,n) connected components which contain at most 2(m+n) vertices altogether. The former bound is tight, the latter is tight up to an additive constant. We also present an optimal O(m+n) time algorithm for computing the bisector.

In planar location problems with barriers one considers regions which are forbidden for the siting of new facilities as well as for trespassing. These problems areimportant since they reflect various real-world situations.The resulting mathematical models have a non-convex objectivefunction and are therefore difficult to tackle using standardmethods of location theory even in the case of simple barriershapes and distance funtions.For the case of center objectives with barrier distancesobtained from the rectilinear or Manhattan metric it is shown that the problem can be solved by identifying a finitedominating set (FDS) the cardinality of which is bounded bya polynomial in the size of the problem input. The resultinggenuinely polynomial algorithm can be combined with bound computations which are derived from solving closely connectedrestricted location and network location problems.It is shown that the results can be extended to barrier center problems with respect to arbitrary block norms having fourfundamental directions.

In this paper we deal with an NP-hard combinatorial optimization problem, the k-cardinality tree problem in node weighted graphs. This problem has several applications , which justify the need for efficient methods to obtain good solutions. We review existing literature on the problem. Then we prove that under the condition that the graph contains exactly one trough, the problem can be solved in ploynomial time. For the general NP-hard problem we implemented several local search methods to obtain heuristics solutions, which are qualitatively better than solutions found by constructive heuristics and which require significantly less time than needed to obtain optimal solutions. We used the well known concepts of genetic algorithms and tabu search with useful extensions. We show that all the methods find optimal solutions for the class of graphs containing exactly one trough. The general performance of our methods as compared to other heuristics is illustrated by numerical results.

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.

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.

The problem of finding an optimal location X* minimizing the maximum Euclidean distance to existing facilities is well solved by e.g. the Elzinga-Hearn algorithm. In practical situations X* will however often not be feasible. We therefore suggest in this note a polynomial algorithm which will find an optimal location X^F in a feasible subset F of the plane R^2

In the following, we discuss a procedure for interpolating a spatial-temporal stochastic process. We stick to a particular, moderately general model but the approach can be easily transered to other similar problems. The original data, which motivated this work, are measurements of gas concentrations (SO2, NO, O2) and several meteorological parameters (temperature, sun radiation, procipitation, wind speed etc.). These date have been and are still recorded twice every hour at several irregularly located places in the forests of the state Rheinland-Pfalz as part of a program monitoring the air pollution in the forests.

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.

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.

We present an approach to learning cooperative behavior of agents. Our ap-proach is based on classifying situations with the help of the nearest-neighborrule. In this context, learning amounts to evolving a set of good prototypical sit-uations. With each prototypical situation an action is associated that should beexecuted in that situation. A set of prototypical situation/action pairs togetherwith the nearest-neighbor rule represent the behavior of an agent.We demonstrate the utility of our approach in the light of variants of thewell-known pursuit game. To this end, we present a classification of variantsof the pursuit game, and we report on the results of our approach obtained forvariants regarding several aspects of the classification. A first implementationof our approach that utilizes a genetic algorithm to conduct the search for a setof suitable prototypical situation/action pairs was able to handle many differentvariants.

The common wisdom that goal orderings can be used to improve planning performance is nearly as old as planning itself. During the last decades of research several approaches emerged that computed goal orderings for different planning paradigms, mostly in the area of state-space planning. For partial-order, plan-space planners goal orderings have not been investigated in much detail. Mechanisms developed for statespace planning are not directly applicable because partial-order planners do not have a current (world) state. Further, it is not completely clear how plan-space planners should make use of goal orderings. This paper describes an approach to extract goal orderings to be used by the plan-space planner CAPlan. The extraction of goal orderings is based on the analysis of an extended version of operator graphs which previously have been found useful for the analysis of interactions and recursion of plan-space planners.

Im Rahmen des Sonderforschungsbereichs SFB314, Projekt X9 "Lernen und Analogie in technischen Expertensystemen", wurde die Verwendbarkeit von Techniken des fallbasierten Schliessens in wissens- basierten Systemen untersucht. Als prototypische Anwendungsdomäne wurde die Arbeitsplanerstellung rotationssymmetrischer Werkstücke gewählt. Im vorliegenden Beitrag wird ein Modell der Arbeits- planerstellung unter Berücksichtigung der verschiedenen, bisher als unabhängig behandelten Planungsmethoden beschrieben. Auf der Basis einer modelbasierte Wissensakquistion aus in Unternehmen verfügbaren Arbeitsplänen wird ein Ausschnitt der Arbeitsplanerstellung, die Aufspannplanung, detailliert. Die Anwendbarkeit wurde durch eine prototypische Realisierung nachgewiesen.

Freivalds, Karpinski and Smith [8] explored a special type of learning in the limit: identification of an unknown concept (function) by eliminating (erasing) all but one possible hypothesis (this type of learning is called co-learning). The motivation behind learning by erasing lies in the process of human and automated computer learning: often we can discard incorrect solutions much easier than to come up with the correct one. In Gödel numberings any learnable family can be learned by an erasing strategy. In this paper we concentrate on co-learning minimal programs. We show that co-learning of minimal programs, as originally defined is significantly weaker than learning minimal programs in Gödel numberings. In order to enhance the learning power

We present an approach to automating the selection of search-guiding heuris-tics that control the search conducted by a problem solver. The approach centerson representing problems with feature vectors that are vectors of numerical val-ues. Thus, similarity between problems can be determined by using a distancemeasure on feature vectors. Given a database of problems, each problem beingassociated with the heuristic that was used to solve it, heuristics to be employedto solve a novel problem are suggested in correspondence with the similaritybetween the novel problem and problems of the database.Our approach is strongly connected with instance-based learning and nearest-neighbor classification and therefore possesses incremental learning capabilities.In experimental studies it has proven to be a viable tool for achieving the finaland crucial missing piece of automation of problem solving - namely selecting anappropriate search-guiding heuristic - in a flexible way.This work was supported by the Deutsche Forschungsgemeinschaft (DFG).

This report presents the properties of a specification of the domain of process planning for rotary symmetrical workpieces. The specification results from a model for problem solving in this domain that involves different reasoners, one of which is an AI planner that achieves goals corresponding to machining workpieces by considering certain operational restrictions of the domain. When planning with SNLP (McAllester and Rosenblitt, 1991), we will show that the resulting plans have the property of minimizing the use of certain key operations. Further, we will show that, for elastic protected plans (Kambhampati et al., 1996) such as the ones produced by SNLP, the goals corresponding to machining parts of a workpiece are OE-constrained trivial serializable, a special form of trivial serializability (Barrett and Weld, 1994). However, we will show that planning with SNLP in this domain can be very difficult: elastic protected plans for machining parts of a workpiece are nonmergeable. Finally, we will show that, for sufix, prefix or sufix and prefix plans such as the ones produced by state-space planners, it is not possible to have both properties, being OEconstrained trivial serializable and minimizing the use of the key operations, at the same time.

In nebenläufigen Systemen erleichtert das Konzept der Atomarität vonOperationen, konkurrierende Zugriffe in größere, leichter beherrschbareAbschnitte zu unterteilen. Wenn wir aber Spezifikationen in der forma-len Beschreibungstechnik Estelle betrachten, erweist es sich, daß es un-ter bestimmten Umständen schwierig ist, die Atomarität der sogenanntenTransitionen bei Implementationen exakt einzuhalten, obwohl diese Ato-marität eine konzeptuelle Grundlage der Semantik von Estelle ist. Es wirdaufgezeigt, wie trotzdem sowohl korrekte als auch effiziente nebenläufigeImplementationen erreicht werden können. Schließlich wird darauf hinge-wiesen, daß die das Problem auslösenden Aktionen oft vom Spezifiziererleicht von vorneherein vermieden werden können; und dies gilt auch überden Kontext von Estelle hinaus.

Top-down and bottom-up theorem proving approaches have each specific ad-vantages and disadvantages. Bottom-up provers profit from strong redundancycontrol and suffer from the lack of goal-orientation, whereas top-down provers aregoal-oriented but have weak calculi when their proof lengths are considered. Inorder to integrate both approaches our method is to achieve cooperation betweena top-down and a bottom-up prover: The top-down prover generates subgoalclauses, then they are processed by a bottom-up prover. We discuss theoreticaspects of this methodology and we introduce techniques for a relevancy-basedfiltering of generated subgoal clauses. Experiments with a model eliminationand a superposition-based prover reveal the high potential of our cooperation approach.The author was supported by the Deutsche Forschungsgemeinschaft (DFG).

We examine an approach for demand-driven cooperative theorem proving.We briefly point out the problems arising from the use of common success-driven cooperation methods, and we propose the application of our approachof requirement-based cooperative theorem proving. This approach allows for abetter orientation on current needs of provers in comparison with conventional co-operation concepts. We introduce an abstract framework for requirement-basedcooperation and describe two instantiations of it: Requirement-based exchangeof facts and sub-problem division and transfer via requests. Finally, we reporton experimental studies conducted in the areas superposition and unfailing com-pletion.The author was supported by the Deutsche Forschungsgemeinschaft (DFG).

HOT is an automated higher-order theorem prover based on HTE, an extensional higher-order tableaux calculus (Kohlhase 95). The first part of the paper introduces a variant of the calculus which closely corresponds to the proof procedure implemented in HOT. The second part discusses HOT's design that can be characterized as a concurrent Blackboard architecture. We show the usefulness of the implementation by including benchmark results for over one hundred solved problems from logic and set theory.

Orderings on polynomial interpretations of operators represent a powerful technique for proving thetermination of rewriting systems. One of the main problems of polynomial orderings concerns thechoice of the right interpretation for a given rewriting system. It is very difficult to develop techniquesfor solving this problem. Here, we present three new heuristic approaches: (i) guidelines for dealingwith special classes of rewriting systems, (ii) an algorithm for choosing appropriate special polynomialsas well as (iii) an extension of the original polynomial ordering which supports the generation ofsuitable interpretations. All these heuristics will be applied to examples in order to illustrate theirpractical relevance.

The problem of providing connectivity for a collection of applications is largely one of data integration: the communicating parties must agree on thesemantics and syntax of the data being exchanged. In earlier papers [#!mp:jsc1!#,#!sg:BSG1!#], it was proposed that dictionaries of definitions foroperators, functions, and symbolic constants can effectively address the problem of semantic data integration. In this paper we extend that earlier work todiscuss the important issues in data integration at the syntactic level and propose a set of solutions that are both general, supporting a wide range of dataobjects with typing information, and efficient, supporting fast transmission and parsing.

Chains of Recurrences (CRs) are a tool for expediting the evaluation of elementary expressions over regular grids. CR based evaluations of elementaryexpressions consist of 3 major stages: CR construction, simplification, and evaluation. This paper addresses CR simplifications. The goal of CRsimplifications is to manipulate a CR such that the resulting expression is more efficiently to evaluate. We develop CR simplification strategies which takethe computational context of CR evaluations into account. Realizing that it is infeasible to always optimally simplify a CR expression, we give heuristicstrategies which, in most cases, result in a optimal, or close-to-optimal expressions. The motivations behind our proposed strategies are discussed and theresults are illustrated by various examples.

Algorithms in Singular
(1999)

A large set of criteria to evaluate formal methods for reactive systems is presented. To make this set more comprehensible, it is structured according to a Concept-Model of formal methods. It is made clear that it is necessary to make the catalogue more specific before applying it. Some of the steps needed to do so are explained. As an example the catalogue is applied within the context of the application domain building automation systems to three different formal methods: SDL, statecharts, and a temporallogic.

Today, the worlds and terminologies of mechanical engineering and software engineering coexist, but they do not always work together seamlessly. Both worlds have developed their own separate formal vocabulary for expressing their concepts as well as for capturing and communicating their respective domain knowledge. But, these two vocabularies are not unified, interwoven, or at least interconnected in a reasonable manner. Thus, the subject of this paper is a comparison of the vocabularies of the two fields, namely feature technology from the area of mechanical engineering and software design patterns from the software engineering domain. Therefore, a certain amount of definitions, history, examples, etc. is presented for features as well as for design patterns. After this, an analysis is carried out to identify analogies and differences. The main intention of this paper is to inform both worlds - mechanical and software engineering - about the other side's terminology and to start a discussion about potential mutual benefits and possibilities to bridge the gap between these two worlds, e.g. to improve the manageability of CAx product development processes.