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Spectral Functions and Smoothing Techniques on Jordan Algebras: How algebraic techniques can help to design efficient optimizati,Used
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Convex optimization has witnessed a considerable progress, mainly due to the development of powerful algorithms and software. In fact, one class of convex problems, called selfscaled, can be particularly efficiently solved. This class encompasses a large amount of reallife convex optimization problems, including linear and semidefinite programming. This class is best described using an algebraic structure known as formally real (or Euclidean) Jordan algebra, which provides an elegant and powerful unifying framework for its study. This book proposes an extensive and selfcontained description of these algebras. Our work focuses on the socalled spectral functions on formally real Jordan algebras, a natural generalization of spectral functions of symmetric matrices. Based on an original variational analysis of eigenvalues in Jordan algebras, we discuss their most important properties, such as differentiability and convexity. We show how these results can be applied to extend several algorithms existing for linear or secondorder programming to the general class of selfscaled problems, e.g. the powerful smoothing techniques of Nesterov.
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