An Introduction to Formally Real Jordan Algebras and Their Applications in Optimization

F. Alizadeh ()
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F. Alizadeh: Rutgers-State University of New Jersey

Chapter Chapter 11 in Handbook on Semidefinite, Conic and Polynomial Optimization, 2012, pp 297-337 from Springer

Abstract: Abstract In this chapter we study formally real Jordan algebras and their impact on certain convex optimization problems. We first show how common topics in convex optimization problems, such as complementarity and interior point algorithms, give rise to algebraic questions. Next we study the basic properties of formally real Jordan algebras including properties of their multiplication operator, quadratic representation, spectral properties and Peirce decomposition. Finally we show how this theory transparently unifies presentation and analysis of issues such as degeneracy and complementarity, and proofs of polynomial time convergence of interior point methods in linear, second order and semidefinite optimization problems.

Keywords: Jordan Algebra; Interior Point Method; Regular Element; Semidefinite Programming; Symmetric Cone (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1007/978-1-4614-0769-0_11

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