On cones of nonnegative quadratic functions
Mathematics of Operations Research
Binary Partitioning, Perceptual Grouping, and Restoration with Semidefinite Programming
IEEE Transactions on Pattern Analysis and Machine Intelligence
A Semidefinite Programming Based Polyhedral Cut and Price Approach for the Maxcut Problem
Computational Optimization and Applications
Semidefinite spectral clustering
Pattern Recognition
Necessary and sufficient global optimality conditions for NLP reformulations of linear SDP problems
Journal of Global Optimization
Rank-constrained separable semidefinite programming for optimal beamforming design
ISIT'09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 4
Rank-constrained separable semidefinite programming with applications to optimal beamforming
IEEE Transactions on Signal Processing
Semidefinite relaxation bounds for bi-quadratic optimization problems with quadratic constraints
Journal of Global Optimization
Active multiple kernel learning for interactive 3D object retrieval systems
ACM Transactions on Interactive Intelligent Systems (TiiS)
Revealing network communities with a nonlinear programming method
Information Sciences: an International Journal
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We derive some basic results on the geometry of semidefinite programming (SDP) and eigenvalue-optimization, i.e., the minimization of the sum of the k largest eigenvalues of a smooth matrix-valued function. We provide upper bounds on the rank of extreme matrices in SDPs, and the first theoretically solid explanation of a phenomenon of intrinsic interest in eigenvalue-optimization. In the spectrum of an optimal matrix, the kth and (k + 1)st largest eigenvalues tend to be equal and frequently have multiplicity greater than two. This clustering is intuitively plausible and has been observed as early as 1975. When the matrix-valued function is affine, we prove that clustering must occur at extreme points of the set of optimal solutions, if the number of variables is sufficiently large. We also give a lower bound on the multiplicity of the critical eigenvalue. These results generalize to the case of a general matrix-valued function under appropriate conditions.