Cluster analysis and mathematical programming

Hansen, Pierre; Jaumard, Brigitte

doi:10.1007/bf02614317

Cited by 401 publications

(254 citation statements)

References 82 publications

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“…The feasibility problem has been studied for other types of constraints or measures of quality [Hansen and Jaumard 1997]. For example, the clustering problem where the quality is measured by the maximum cluster diameter can be transformed in an obvious way into a constrained clustering problem.…”

Section: Preliminary Definitions and Terminologymentioning

confidence: 99%

The complexity of non-hierarchical clustering with instance and cluster level constraints

Davidson

Ravi

2007

Data Min Knowl Disc

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Recent work has looked at extending clustering algorithms with instance level must-link (ML) and cannot-link (CL) background information. Our work introduces δ and ǫ cluster level constraints that influence inter-cluster distances and cluster composition. The addition of background information, though useful at providing better clustering results, raises the important feasibility question: Given a collection of constraints and a set of data, does there exist at least one partition of the data set satisfying all the constraints? We study the complexity of the feasibility problem for each of the above constraints separately and also for combinations of constraints. Our results clearly delineate combinations of constraints for which the feasibility problem is computationally intractable (i.e., NP-complete) from those for which the problem is efficiently solvable (i.e., in the computational class P).We also consider the ML and CL constraints in conjunctive and disjunctive normal forms (CNF and DNF respectively). We show that for ML constraints, the feasibility problem is intractable for CNF but efficiently solvable for DNF. Unfortunately, for CL constraints, the feasibility problem is intractable for both CNF and DNF. This effectively means that CLconstraints in a non-trivial form cannot be efficiently incorporated into clustering algorithms. To overcome this, we introduce the notion of a choice-set of constraints and prove that the feasibility problem for choice-sets is efficiently solvable for both ML and CL constraints. We also present empirical results which indicate that the feasibility problem occurs extensively in real world problems.

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Section: Preliminary Definitions and Terminologymentioning

confidence: 99%

The complexity of non-hierarchical clustering with instance and cluster level constraints

Davidson

Ravi

2007

Data Min Knowl Disc

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“…Currently the complexity status of this problem is open, in spite of a number of attempts to prove that this problem is NP-hard (see the survey [1]). This problem has been deeply studied during the last 50 years and a number of exact and heuristic approaches to its solution have been developed (see the survey in [11]). Some of them will be discussed in Section 4.…”

Section: Proposition 1 Suppose W It = W I Does Not Depend On T For mentioning

confidence: 99%

“…With simplifying assumption that the prices are stationary distributed, the estimated variancê σ 2 i of price c it in node i, as well as the estimated correlationρ ij between the nodal price c it and the hub price c j t , can be expressed on the basis of the historical data. Then equation (11) leads to the following criterion:…”

Section: Proposition 1 Suppose W It = W I Does Not Depend On T For mentioning

confidence: 99%

“…A number of other exact and heuristic approaches, such as the branch-and-bound algorithms, metaheuristics, hierarchical clustering heuristics and other methods (see e.g. [11,12] and references there) can be also applied to the Hubs Construction Problem and its modifications. However, the main limiting factor, which may hinder the usage of some of these methods, is a large dimensionality of typical instances of the Hubs Construction Problem.…”

Section: The K-means Heuristicmentioning

confidence: 99%

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Trading Hubs Construction for Electricity Markets

Borisovsky

Eremeev

Grinkevich³

et al. 2009

Optimization in the Energy Industry

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In this chapter, we consider a problem of constructing trading hubs in the structure of the electricity wholesale markets. The nodes of a trading hub are used to calculate a reference price that can be employed by the market participants for different types of hedging. The need for such a reference price is due to considerable variability of energy prices at different nodes of the electricity grid at different periods of time. The hubs construction is viewed as a mathematical programming problem here. We discuss its connections with clustering problems, consider the heuristic algorithms of solution and indicate some complexity issues. The performance of algorithms is illustrated on the real-life data.

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“…Hartigan [28]; Jain, Murty and Flynn [31]; Mirkin [44]). Many different criteria are used in the literature to express homogeneity and/or separation of the clusters to be found (see [22] for a survey). Among them, a frequently used one is the minimum sum of squared Euclidean distances from each entity to the centroid of the cluster to which it belongs.…”

Section: Introductionmentioning

confidence: 99%

An improved column generation algorithm for minimum sum-of-squares clustering

2010

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Given a set of entities associated with points in Euclidean space, minimum sum-of-squares clustering (MSSC) consist in partitioning this set into clusters such that the sum of squared distances from each point to the centroid of its cluster is minimized. A column generation algorithm for MSSC was given in du Merle et al. [15]. The bottleneck of that algorithm is resolution of the auxiliary problem of finding a column with negative reduced cost. We propose a new way to solve this auxiliary problem based on geometric arguments. This greatly improves the efficiency of the whole algorithm and leads to exact solution of instances with over 2300 entities, i.e., more than 10 times as much as previously done.

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Cluster analysis and mathematical programming

Cited by 401 publications

References 82 publications

The complexity of non-hierarchical clustering with instance and cluster level constraints

The complexity of non-hierarchical clustering with instance and cluster level constraints

Trading Hubs Construction for Electricity Markets

An improved column generation algorithm for minimum sum-of-squares clustering

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