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Linear systems in Jordan algebras and primal-dual interior-point algorithms. (English) Zbl 0889.65066
The author discusses a possibility of the extension of a primal-dual interior-point algorithm for optimization problems of the form $\langle a,x\rangle \rightarrow \min X \in (b+X) \cap \Omega,$ where $$V$$ is an Euclidean Jordan algebra, $$X$$ is a vector subspace of $$V$$, $$\Omega$$ is the cone of squares of $$V$$, $$a\in X, b \in X^\perp$$. The question of solvability of a linear system arising in the implementation of the primal-dual algorithm is analyzed and a nondegeneracy theory for the considered class of problems is developed.

MSC:
 65K05 Numerical mathematical programming methods 90C05 Linear programming
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References:
 [1] Alizadeh, F.; Haeberly, J.-P.A.; Overton, M.L., Primal-dual interior-point methods for semidefinite programming, (1994), Manuscript · Zbl 0819.65098 [2] Alizadeh, F.; Haeberly, J.-P.A.; Overton, M.L., Primal-dual interior-point methods for semidefinite programming: convergence rates, stability and numerical results, (1996), Preprint [3] Alizadeh, F.; Haeberly, J.-P.A.; Overton, M.L., Complementarity and nondegeneracy in semidefinite programming, Math. programming, ser. B, 77, 111-128, (1997) · Zbl 0890.90141 [4] Faraut, J.; Koranyi, A., Analysis on symmetric cones, (1994), Clarendon Press Oxford · Zbl 0841.43002 [5] Faybusovich, L., Jordan algebras, symmetric cones and interior-point methods, () [6] Freudenthal, H., Oktaven, (1951), Ausnahmegruppen und Oktavengeometrie Utrecht · Zbl 0056.25905 [7] Jacobson, N., Structure and representations of Jordan algebras, (1968), Amer. Math. Society Providence, RI · Zbl 0218.17010 [8] Shida, M.; Shindoh, S.; Kojima, M., Existence of search directions in interior-point algorithms for the SDP and the monotone SDLCP, () · Zbl 0913.90252 [9] Unterberger, A.; Upmeier, H., Pseudodifferential analysis on symmetric cones, (1996), CRC Press · Zbl 0854.35001 [10] Upmeier, H., Toeplitz operators and index theory in several complex variables, (1996), Birkhauser Basel · Zbl 0957.47023
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