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

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Handbook on Semidefinite, Conic and Polynomial Optimization

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 166))

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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.

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Notes

  1. 1.

    The Jordan algebra invented by Jordan was actually the formally real version.

  2. 2.

    In general algebras are defined over finite or infinite dimensional vector spaces, but in this chapter we are exclusively concerned with finite dimensional ones.

  3. 3.

    We will write Trace (⋅) and Det (⋅) in the usual matrix theoretic sense, and tr (⋅) and det (⋅) for Jordan algebras.

  4. 4.

    Recall that if \mathop{ vec} (X) is the vector obtained from stacking columns of the matrix X, then we have the identity \mathop{ vec} (ZYX ⊤ ) = (X ⊗ Z)\mathop{ vec} (Y ), see for example [21].

  5. 5.

    Pronounced like “purse”.

  6. 6.

    We write SDP as an abbreviation of semidefinite programming over matrices, and scp for symmetric conic programming.

  7. 7.

    In the remainder of this chapter the order of eigenvalues is not essential. Thus, to avoid writing \({\lambda }_{{\pi }_{i}}\) and \({\omega }_{{\pi }_{i}}\), we will simply write λi and ωi.

  8. 8.

    A linear transformation D on \(\mathbb{E}\) is a derivation if \(D(\mathbf{x} \circ \mathbf{y}) ={\bigl ( D(\mathbf{x})\bigr )} \circ \mathbf{y} +{\bigl ( D(\mathbf{y})\bigr )} \circ \mathbf{x}\) for all \(\mathbf{x},\mathbf{y} \in \mathbb{E}\).

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Acknowledgements

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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  1. Rutgers Business School and RUTCOR, Rutgers-State University of New Jersey, 640 Bartholomew Rd, Piscataway, NJ, 08854, USA

    F. Alizadeh

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  1. Department of Mathematics and Industrial, Ecole Polytechnique Montreal, Édouard-Montpetit Boulevard 2350, Montreal, QC, Canada, H3C 3A7

    Miguel F. Anjos

  2. LAAS-CNRS, Avenue du Colonel Roche 7, 31077, Toulouse Cedex 4, France

    Jean B. Lasserre

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Alizadeh, F. (2012). An Introduction to Formally Real Jordan Algebras and Their Applications in Optimization. In: Anjos, M.F., Lasserre, J.B. (eds) Handbook on Semidefinite, Conic and Polynomial Optimization. International Series in Operations Research & Management Science, vol 166. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0769-0_11

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