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Many computational science and data analysis techniques lead to optimizing Rayleigh-Quotient (RQ) and RQ-type objective functions, such as computing excitation states (energies) of electronic structures, robust classification to handle uncertainty and constrained data clustering to incorporate a prior information. In this talk, we will discuss origins of some RQ optimizations, variational principles, and reformulations to algebraic linear and nonlinear eigenvalue problems. We will show how to exploit underlying properties of eigenvalue problems for designing eigensolvers, and illustrate the efficacy of these solvers in electronic structure calculations and constrained image segmentation.