Applications of Variational Analysis to a Generalized Fermat-Torricelli Problem

In this paper we develop new applications of variational analysis and generalized differentiation to the following optimization problem and its specifications: given n closed subsets of a Banach space, find such a point for which the sum of its distances to these sets is minimal. This problem can be viewed as an extension of the celebrated Fermat-Torricelli problem: given three points on the plane, find another point that minimizes the sum of its distances to the designated points. The generalized Fermat-Torricelli problem formulated and studied in this paper is of undoubted mathematical interest and is promising for various applications including those frequently arising in location science, optimal networks, etc. Based on advanced tools and recent results of variational analysis and generalized differentiation, we derive necessary as well as necessary and sufficient optimality conditions for the extended version of the Fermat-Torricelli problem under consideration, which allow us to completely solve it in some important settings. Furthermore, we develop and justify a numerical algorithm of the subgradient type to find optimal solutions in convex settings and provide its numerical implementations.     

A regularized decomposition method for minimizing a sum of polyhedral functions

A problem of minimizing a sum of many convex piecewise-linear functions is considered. In view of applications to two-stage linear programming, where objectives are marginal values of lower level problems, it is assumed that domains of objectives may be proper polyhedral subsets of the space of decision variables and are defined by piecewise-linear induced feasibility constraints. We propose a new decomposition method that may start from an arbitrary point and simultaneously processes objective and feasibility cuts for each component. The master program is augmented with a quadratic regularizing term and comprises an a priori bounded number of cuts. The method goes through nonbasic points, in general, and is finitely convergent without any nondegeneracy assumptions. Next, we present a special technique for solving the regularized master problem that uses an active set strategy and QR factorization and exploits the structure of the master. Finally, some numerical evidence is given.On leave from Instytut Automatyki, Politechnika Warszawska, Poland.     

An approximation algorithm for solving a problem of cluster analysis

The authors provide some 2-approximation algorithm for an intractable problem to which one can reduce the problem of partitioning a vector set in Euclidean space into the two subsets (clusters) having the minimum sum of distance squares.     

A Semilinear Structure on Semigroups in a Metric Space

We introduce a sort of semilinear structure on subsets of the family of semigroups defined on a metric space. The key step is the definition of the sum of two semigroups, which is here achieved by means of the classical operator splitting technique. No linearity assumption is required, since the whole construction is in a metric space.     

Exact pseudopolynomial algorithms for a balanced 2-clustering problem

We consider the strongly NP-hard problem of partitioning a set of Euclidean points into two clusters so as to minimize the sum (over both clusters) of the weighted sum of the squared intracluster distances from the elements of the clusters to their centers. The weights of sums are the sizes of the clusters. The center of one cluster is given as input, while the center of the other cluster is unknown and determined as the average value over all points in the cluster (as the geometric center). Two variants of the problems are analyzed in which the cluster sizes are either given or unknown. We present and prove some exact pseudopolynomial algorithms in the case of integer components of the input points and fixed space dimension.     

Duality theorem for a generalized Fermat-Weber problem

The classical Fermat-Weber problem is to minimize the sum of the distances from a point in a plane tok given points in the plane. This problem was generalized by Witzgall ton-dimensional space and to allow for a general norm, not necessarily symmetric; he found a dual for this problem. The authors generalize this result further by proving a duality theorem which includes as special cases a great variety of choices of norms in the terms of the Fermat-Weber sum. The theorem is proved by applying a general duality theorem of Rockafellar. As applications, a dual is found for the multi-facility location problem and a nonlinear dual is obtained for a linear programming problem with a priori bounds for the variables. When the norms concerned are continuously differentiable, formulas are obtained for retrieving the solution for each primal problem from the solution of its dual.     

Polynomial-Time Approximation Algorithm for the Problem of Cardinality-Weighted Variance-Based 2-Clustering with a Given Center

A strongly NP-hard problem of partitioning a finite set of points of Euclidean space into two clusters is considered. The solution criterion is the minimum of the sum (over both clusters) of weighted sums of squared distances from the elements of each cluster to its geometric center. The weights of the sums are equal to the cardinalities of the desired clusters. The center of one cluster is given as input, while the center of the other is unknown and is determined as the point of space equal to the mean of the cluster elements. A version of the problem is analyzed in which the cardinalities of the clusters are given as input. A polynomial-time 2-approximation algorithm for solving the problem is constructed.     

A fully polynomial-time approximation scheme for a sequence 2-cluster partitioning problem

We consider a strongly NP-hard problem of partitioning a finite sequence of points in Euclidean space into the two clustersminimizing the sum over both clusters of intra-cluster sums of squared distances from the clusters elements to their centers. The sizes of the clusters are fixed. The centroid of the first cluster is defined as the mean value of all vectors in the cluster, and the center of the second cluster is given in advance and equals 0. Additionally, the partition must satisfy the restriction that for all vectors in the first cluster the difference between the indices of two consequent points from this cluster is bounded from below and above by some given constants.We present a fully polynomial-time approximation scheme for the case of fixed space dimension.     

Formulations and valid inequalities for the node capacitated graph partitioning problem

We investigate the problem of partitioning the nodes of a graph under capacity restriction on the sum of the node weights in each subset of the partition. The objective is to minimize the sum of the costs of the edges between the subsets of the partition. This problem has a variety of applications, for instance in the design of electronic circuits and devices. We present alternative integer programming formulations for this problem and discuss the links between these formulations. Having chosen to work in the space of edges of the multicut, we investigate the convex hull of incidence vectors of feasible multicuts. In particular, several classes of inequalities are introduced, and their strength and robustness are analyzed as various problem parameters change.     

Fully polynomial-time approximation scheme for a special case of a quadratic Euclidean 2-clustering problem

The strongly NP-hard problem of partitioning a finite set of points of Euclidean space into two clusters of given sizes (cardinalities) minimizing the sum (over both clusters) of the intracluster sums of squared distances from the elements of the clusters to their centers is considered. It is assumed that the center of one of the sought clusters is specified at the desired (arbitrary) point of space (without loss of generality, at the origin), while the center of the other one is unknown and determined as the mean value over all elements of this cluster. It is shown that unless P = NP, there is no fully polynomial-time approximation scheme for this problem, and such a scheme is substantiated in the case of a fixed space dimension.     

A global optimization procedure for the location of a median line in the three-dimensional space

A global optimization procedure is proposed to find a line in the Euclidean three-dimensional space which minimizes the sum of distances to a given finite set of three-dimensional data points.Although we are using similar techniques as for location problems in two dimensions, it is shown that the problem becomes much harder to solve. However, a problem parameterization as well as lower bounds are suggested whereby we succeeded in solving medium-size instances in a reasonable amount of computing time.     

On Polynomial Solvability of One Quadratic Euclidean Clustering Problem on a Line

Doklady Mathematics - We consider the problem of partitioning a finite set of points in Euclidean space into clusters so as to minimize the sum, over all clusters, of the intracluster sums of the...     

局部有限超拓扑的局部紧性

本文讨论了赋予局部有限拓扑的非空闲子集超空间的局部紧性.主要结果是:X正则,则其闭子集超空间局部紧当且仅当X可表示成一个紧空间与一个离散空间的拓扑和.     

Variable neighborhood search for minimum sum-of-squares clustering on networks

Euclidean Minimum Sum-of-Squares Clustering amounts to finding p prototypes by minimizing the sum of the squared Euclidean distances from a set of points to their closest prototype. In recent years related clustering problems have been extensively analyzed under the assumption that the space is a network, and not any more the Euclidean space. This allows one to properly address community detection problems, of significant relevance in diverse phenomena in biological, technological and social systems. However, the problem of minimizing the sum of squared distances on networks have not yet been addressed. Two versions of the problem are possible: either the p prototypes are sought among the set of nodes of the network, or also points along edges are taken into account as possible prototypes. While the first problem is transformed into a classical discrete p-median problem, the latter is new in the literature, and solved in this paper with the Variable Neighborhood Search heuristic. The solutions of the two problems are compared in a series of test examples.     

On a robustness property in single-facility location in continuous space

We consider a single-facility location problem in continuous space—here the problem of minimizing a sum or the maximum of the possibly weighted distances from a facility to a set of points of demand. The main result of this paper shows that every solution (optimal facility location) of this problem has an interesting robustness property. Any optimal facility location is the most robust in the following sense: given a suitable highest admissible cost, it allows the greatest perturbation of the locations of the demand without exceeding this highest admissible chosen cost.     

On the subset sum problem for finite fields

Let G be the additive group of a finite field. J. Li and D. Wan determined the exact number of solutions of the subset sum problem over G, by giving an explicit formula for the number of subsets of G of prescribed size whose elements sum up to a given element of G. They also determined a closed-form expression for the case where the subsets are required to contain only nonzero elements. In this paper we give an alternative proof of the two formulas. Our argument is purely combinatorial, as in the original proof by Li and Wan, but follows a different and somehow more “natural” approach. We also indicate some new connections with coding theory and combinatorial designs.     

An Approximation Scheme for the Problem of Finding a Subsequence

We consider a strongly NP-hard Euclidean problem of finding a subsequence in a finite sequence. The criterion of the solution is the minimum sum of squared distances from the elements of a sought subsequence to its geometric center (centroid). It is assumed that a sought subsequence contains a given number of elements. In addition, a sought subsequence should satisfy the following condition: the difference between the indices of each previous and subsequent points is bounded with given lower and upper constants.We present an approximation algorithm of solving the problem and prove that it is a fully polynomial-time approximation scheme (FPTAS) when the space dimension is bounded by a constant.     

Robust registration of surfaces using a refined iterative closest point algorithm with a trust region approach

The problem of finding a rigid body transformation, which aligns a set of data points with a given surface, using a robust M-estimation technique is considered. A refined iterative closest point (ICP) algorithm is described where a minimization problem of point-to-plane distances with a proposed constraint is solved in each iteration to find an updating transformation. The constraint is derived from a sum of weighted squared point-to-point distances and forms a natural trust region, which ensures convergence. Only a minor number of additional computations are required to use it. Two alternative trust regions are introduced and analyzed. Finally, numerical results for some test problems are presented. It is obvious from these results that there is a significant advantage, with respect to convergence rate of accuracy, to use the proposed trust region approach in comparison with using point-to-point distance minimization as well as using point-to-plane distance minimization and a Newton- type update without any step size control.     

On the Complexity of Some Problems of Searching for a Family of Disjoint Clusters

Doklady Mathematics - Two consimilar problems of searching for a family of disjoint subsets (clusters) in a finite set of points of Euclidean space are considered. In these problems, the task is to...     

A duality theorem for a special class of programming problems in complex space

Duality relations for the programming problem of a special class where the objective function is a sum of positive-semidefinite quadratic forms, and a sum of square roots of positive-semidefinite quadratic forms, over a convex polyhedral cone in complex space are considered. The duality relations between the primal problem and its dual are established.