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Design and Analysis of Optimization Algorithms for Solving Nonlinear Optimization Pproblems with Orthogonal Constraints and Certain Applications

Dehghanpour, Jafar | 2024

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 56804 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Mahdavi Amiri, Nezamoddin
  7. Abstract:
  8. 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 smooth nonlinear programming problem with quadratic (convex) constraints that can be solved by a proper optimization method . We first make use of an optimization method with two gradient projection steps and then apply a post-processing technique to construct a partition of the clustering problem . Comparative performance of our proposed approach with other available clustering methods over randomly generated test problems and hard synthetic data-sets shows the outperformance of our approach , in terms of the obtained misclassification error rate and the Rand index
  9. Keywords:
  10. Non-Negative Matrix Factorization (NMF) ; Isoperimetry ; Clustering ; Optimization Problems ; Optimization Problem with Orthogonality Constraints ; Orthogonal Nonnegative Matrix Factorization (ONMF)

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