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Orthogonal Nonnegative Matrix Factorization (ONMF) with orthogonality constraints on a matrix has been found to provide better clustering results over existing clustering problems . Because of the orthogonality constraint , this optimization problem is difficult to solve . Many of the existing constraint-preserving methods deal directly with the constraints using different techniques such as matrix decomposition or computing exponential matrices . Here , we propose an alternative formulation of the ONMF problem which converts the orthogonality constraints into non-convex constraints . To handle the non-convex constraints , a penalty function is applied . The penalized problem is a... 

Graph partitioning and clustering are among the most important problems in computer science and engineering with a wide range of applications, to which a large number of research projects have attended. The normalized cut ratio is one of the best metrics for graph partitioning, giving rise to the ``Isoperimetry'' problem, which is, however, a challenging problem in its various forms. In this dissertation, we investigate the Isoperimetry problem. %, focusing on the Mean version. We prove that the Mean Isoperimetry problem on edge-weighted trees is solvable in strongly polynomial time. The algorithm can accomodate a wide range of constraints on the problem in the connected regime, including...