Structured Optimization Problems

Abstract

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.