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.
Knowledge about the distribution of a statistical estimator is important for various purposes like, for example, the construction of confidence intervals for model parameters or the determiation of critical values of tests. A widely used method to estimate this distribution is the so-called bootstrap which is based on an imitation of the probabilistic structure of the data generating process on the basis of the information provided by a given set of random observations. In this paper we investigate this classical method in the context of artificial neural networks used for estimating a mapping from input to output space. We establish consistency results for bootstrap estimates of the distribution of parameter estimates.
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.
In this paper a group of participants of the 12th European Summer Institute which took place in Tenerifa, Spain in June 1995 present their views on the state of the art and the future trends in Locational Analysis. The issue discussed includes modelling aspects in discrete, network and continuous location, heuristic techniques, the state of technology and undesirable facility location. Some general questions are stated reagrding the applicability of location models, promising research directions and the way technology affects the development of solution techniques.
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.
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.