A Class of Parametric Tree-Based Clustering Methods

We introduce a class of tree-based clustering methods based on a single parameter W and show how to generate the full collection of cluster sets C(W), without duplication, by varying W according to conditions identified during the algorithm’s execution. The number of clusters within C(W) for a given W is determined automatically, using a graph representation in which cluster elements are represented by nodes and their pairwise connections are represented by edges. We identify features of the clusters produced which lead to special procedures to accelerate the computation. Finally, we introduce a related node-based variant of the algorithm based on a parameter Y which can be used to generate clusters with complementary features, and a method that combines both variants based on a parameter Z and a weight that determines the contribution of each variant

Higher-order cover cuts from zero–one knapsack constraints augmented by two-sided bounding inequalities

AbstractExtending our work on second-order cover cuts [F. Glover, H.D. Sherali, Second-order cover cuts, Mathematical Programming (ISSN: 0025-5610 1436-4646) (2007), doi:10.1007/s10107-007-0098-4. (Online)], we introduce a new class of higher-order cover cuts that are derived from the implications of a knapsack constraint in concert with supplementary two-sided inequalities that bound the sums of sets of variables. The new cuts can be appreciably stronger than the second-order cuts, which in turn dominate the classical knapsack cover inequalities. The process of generating these cuts makes it possible to sequentially utilize the second-order cuts by embedding them in systems that define the inequalities from which the higher-order cover cuts are derived. We characterize properties of these cuts, design specialized procedures to generate them, and establish associated dominance relationships. These results are used to devise an algorithm that generates all non-dominated higher-order cover cuts, and, in particular, to formulate and solve suitable separation problems for deriving a higher-order cut that deletes a given fractional solution to an underlying continuous relaxation. We also discuss a lifting procedure for further tightening any generated cut, and establish its polynomial-time operation for unit-coefficient cuts. A numerical example is presented that illustrates these procedures and the relative strength of the generated non-redundant, non-dominated higher-order cuts, all of which turn out to be facet-defining for this example. Some preliminary computational results are also presented to demonstrate the efficacy of these cuts in comparison with lifted minimal cover inequalities for the underlying knapsack polytope