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

  • F. AlizadehEmail author
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 166)


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.


Jordan Algebra Interior Point Method Regular Element Semidefinite Programming Symmetric Cone 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The author greatly appreciates the thorough reading of this chapter by an anonymous reviewer, and correcting numerous stylistic and some mathematical errors present in earlier versions. This work was supported in part by the US National Science Foundation Grant number CMMI-0935305.


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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Rutgers Business School and RUTCORRutgers-State University of New JerseyPiscatawayUSA

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