Guaranteed clustering and biclustering via semidefinite programming

Identifying clusters of similar objects in data plays a significant role in a wide range of applications. As a model problem for clustering, we consider the densest k-disjoint-clique problem, whose goal is to identify the collection of k disjoint cliques of a given weighted complete graph maximizing the sum of the densities of the complete subgraphs induced by these cliques. In this paper, we establish conditions ensuring exact recovery of the densest k cliques of a given graph from the optimal solution of a particular semidefinite program. In particular, the semidefinite relaxation is exact for input graphs corresponding to data consisting of k large, distinct clusters and a smaller number of outliers. This approach also yields a semidefinite relaxation for the biclustering problem with similar recovery guarantees. Given a set of objects and a set of features exhibited by these objects, biclustering seeks to simultaneously group the objects and features according to their expression levels. This problem may be posed as partitioning the nodes of a weighted bipartite complete graph such that the sum of the densities of the resulting bipartite complete subgraphs is maximized. As in our analysis of the densest k-disjoint-clique problem, we show that the correct partition of the objects and features can be recovered from the optimal solution of a semidefinite program in the case that the given data consists of several disjoint sets of objects exhibiting similar features. Empirical evidence from numerical experiments supporting these theoretical guarantees is also provided

Immobile indices and CQ-free optimality criteria for linear copositive programming problems

We consider problems of linear copositive programming where feasible sets consist of vectors for which the quadratic forms induced by the corresponding linear matrix combinations are nonnegative over the nonnegative orthant. Given a linear copositive problem, we define immobile indices of its constraints and a normalized immobile index set. We prove that the normalized immobile index set is either empty or can be represented as a union of a finite number of convex closed bounded polyhedra. We show that the study of the structure of this set and the connected properties of the feasible set permits to obtain new optimality criteria for copositive problems. These criteria do not require the fulfillment of any additional conditions (constraint qualifications or other). An illustrative example shows that the optimality conditions formulated in the paper permit to detect the optimality of feasible solutions for which the known sufficient optimality conditions are not able to do this. We apply the approach based on the notion of immobile indices to obtain new formulations of regularized primal and dual problems which are explicit and guarantee strong duality.publishe

Reversible cerebral venulitis in a patient with neuro-Behçet disease

PubMed ID: 22805643[No abstract available

Probabilistic choice models for product pricing based on reservation prices

We consider revenue management models for pricing a product line with several customer segments, working under the assumption that every customer’s product choice is determined entirely by their reservation price. We model the customer choice behavior by several probabilistic choice models and formulate the problems as mixed-integer programming problems. We study special properties of these formulations and compare the resulting optimal prices of the different probabilistic choice models. We also explore some heuristics and valid inequalities to improve the running time of the mixed-integer programming problems. We illustrate the computational results of our models on real and generated customer data taken from a company in the tourism sector