Structured Optimization Problems

Part of the Graduate Texts in Mathematics book series (GTM, volume 258)


In this chapter, we solve several important well-known problems using optimization techniques. These include the extensive theories of the eigenvalues of symmetric matrices and the singular values of a matrix, an optimization problem in Broyden's method for solving nonlinear systems of equations, an optimization problem appearing in quasi-Newton methods for unconstrained minimization of a nonlinear function, the inequalities of Kantorovich, Hadamard, and Hilbert, and the problem of inscribing a maximum-volume ellipsoid in a convex polytope in \( \mathbb{R}^n \). The variational approach to the eigenvalues and singular values are especially important, both in finite and infinite dimensions, since they can be used to prove various inequalities among the eigenvalues (and the singular values), and to establish the spectral decomposition of compact operators in Hilbert spaces, for example.


Spectral Decomposition Multiplier Vector Structure Optimization Problem Simultaneous Diagonalization Equivalent Minimization Problem 
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Copyright information

© Springer New York 2010

Authors and Affiliations

  1. 1.Department of Mathematics & StatisticsUniversity of MarylandBaltimoreUSA

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