John Gunnar Carlsson

Extracting insights from the shape of complex data using topology

This paper applies topological methods to study complex high dimensional data sets by extracting shapes (patterns) and obtaining insights about them. Our method combines the best features of existing standard methodologies such as principal component and cluster analyses to provide a geometric representation of complex data sets. Through this hybrid method, we often find subgroups in data sets that traditional methodologies fail to find. Our method also permits the analysis of individual data sets as well as the analysis of relationships between related data sets. We illustrate the use of our method by applying it to three very different kinds of data, namely gene expression from breast tumors, voting data from the United States House of Representatives and player performance data from the NBA, in each case finding stratifications of the data which are more refined than those produced by standard methods.

Dividing a Territory Among Several Vehicles

We consider an uncapacitated stochastic vehicle routing problem in which vehicle depot locations are fixed, and client locations in a service region are unknown but are assumed to be independent and identically distributed samples from a given probability density function. We present an algorithm for partitioning the service region into subregions so as to balance the workloads of all vehicles when the service region is simply connected and point-to-point distances follow some “natural” metric, such as any Lp norm. This algorithm can also be applied to load balancing of other combinatorial structures, such as minimum spanning trees and minimum matchings.

A Review of Piecewise Linearization Methods

Various optimization problems in engineering and management are formulated as nonlinear programming problems. Because of the nonconvexity nature of this kind of problems, no efficient approach is available to derive the global optimum of the problems. How to locate a global optimal solution of a nonlinear programming problem is an important issue in optimization theory. In the last few decades, piecewise linearization methods have been widely applied to convert a nonlinear programming problem into a linear programming problem or a mixed-integer convex programming problem for obtaining an approximated global optimal solution. In the transformation process, extra binary variables, continuous variables, and constraints are introduced to reformulate the original problem. These extra variables and constraints mainly determine the solution efficiency of the converted problem. This study therefore provides a review of piecewise linearization methods and analyzes the computational efficiency of various piecewise linearization methods.

Robust Partitioning for Stochastic Multivehicle Routing

The problem of coordinating a fleet of vehicles so that all demand points on a territory are serviced and the workload is most evenly distributed among the vehicles is a hard one. For this reason, it is often an effective strategy to first divide the service region and impose that each vehicle is only responsible for its own subregion. This heuristic also has the practical advantage that over time, drivers become more effective at serving their territory and customers. In this paper, we assume that client locations are unknown at the time of partitioning the territory and that each of them will be drawn identically and independently according to a distribution that is actually also unknown. In practice, it might be impossible to identify precisely the distribution if, for instance, information about the demand is limited to historical data. Our approach suggests partitioning the region with respect to the worst-case distribution that satisfies first- and second-order moments information. As a side product...

Finding equitable convex partitions of points in a polygon efficiently

Previous work has developed algorithms for finding an equitable convex partition that partitions the plane into n convex pieces each containing an equal number of red and blue points. Motivated by a vehicle routing heuristic, we look at a related problem where each piece must contain one point and an equal fraction of the area of some convex polygon. We first show how algorithms for solving the older problem lead to approximate solutions for this new equitable convex partition problem. Then we demonstrate a new algorithm that finds an exact solution to our problem in O(N nlog N) time or operations, where n is the number of points, m the number of vertices or edges of the polygon, and N:=n+m the sum.

Dividing a Territory Among Several Facilities

We consider the problem of dividing a geographic region into subregions so as to minimize the maximum workload of a collection of facilities over that region. We assume that the cost of servicing a demand point is a monomial function of the distance to its assigned facility and that demand points follow a continuous probability density. We show that, when our objective is to minimize the maximum workload of all facilities, the optimal partition consists of a collection of circular arcs that are induced by a multiplicatively weighted Voronoi diagram. When we require that all subregions have equal area, the optimal partition consists of a collection of hyperbolic or quartic curves. We show that, for both problems, the dual variables correspond to “prices” for a facility to serve a demand point, and our objective is to determine a set of prices such that the entire region is “purchased” by the facilities, i.e., that the market clears. This allows us to solve the partitioning problem quickly without discretizing the service region.

Euclidean Hub-and-Spoke Networks

The hub-and-spoke distribution paradigm has been a fundamental principle in geographic network design for more than 40 years. One of the primary advantages that such networks possess is their ability to exploit economies of scale in transportation by aggregating network flows through common sources. In this paper, we consider the problem of designing an optimal hub-and-spoke network in continuous Euclidean space: the “spokes” of the network are distributed uniformly over a service region, and our objective is to determine the optimal number of hub nodes and their locations. We consider seven different backbone network topologies for connecting the hub nodes, namely, the Steiner and minimum spanning trees, a travelling salesman tour, a star network, a capacitated vehicle routing tour, a complete bipartite graph, and a complete graph. We also perform an additional analysis on a multilevel network in which network flows move through multiple levels of transshipment before reaching the service region. We desc...

Optimization Theory, Methods, and Applications in Engineering 2014

1 Department of Business Management, National Taipei University of Technology, No. 1, Section 3, Chung-Hsiao East Road, Taipei 10608, Taiwan 2 Program in Industrial and Systems Engineering, University of Minnesota, 111 Church Street SE, Minneapolis, MN 55455, USA 3 Antai College of Economics & Management, Shanghai Jiao Tong University, Room 423, North Building, 535 Fahua Zhen Road, Changning, Shanghai 200052, China 4 Department of Business Administration, Chung Yuan Christian University, Chung-Li 320, Taiwan 5 Department of Information and Electronic Engineering, Muroran Institute of Technology, 27-1 Mizumoto-cho, Muroran 050-8585, Japan

Continuous Facility Location with Backbone Network Costs

We consider a continuous facility location problem in which our objective is to minimize the weighted sum of three costs: 1 fixed costs from installing the facilities, 2 backbone network costs incurred from connecting the facilities to each other, and 3 transportation costs incurred from providing services from the facilities to the service region. We first analyze the limiting behavior of this model and derive the two asymptotically optimal configurations of facilities: one of these configurations is the well studied honeycomb heuristic, and the other is an Archimedean spiral. We then give a fast constant-factor approximation algorithm for finding the placement of a set of facilities in any convex polygon that minimizes the sum of the three aforementioned costs.

Balancing workloads of service vehicles over a geographic territory

Autonomous vehicles (or drones) are very frequently used for servicing a geographic region in numerous applications. Given a geographic territory and a set of n fixed vehicle depots, we consider the problem of designing service districts so as to balance the workload of a collection of vehicles which service this region. We assume that the territory is a connected polygonal region, i.e. a simply connected polygon containing a set of simply connected obstacles. We give a fast algorithm, based on an infinite-dimensional optimization formulation, that divides the territory into compact, connected sub-regions, each of which contains a vehicle depot, such that all regions have equal area. We also show how we can use this algorithm to find better locations of the vehicle depots.