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Keywords
(11)
Automorphism Group
Euclidean Jordan Algebra
Exponential Map
Jordan Algebra
Newton Method
Optimization Problem
Polar Decomposition
Second Order Cone Program
semidefinite program
Interior Point
Interior Point Method
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The
The,10.1007/s109570109777z,Journal of Optimization Theory and Applications,Farid Alizadeh,Yu Xia
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Farid Alizadeh
,
Yu Xia
The Q method of semidefinite programming, developed by Alizadeh, Haeberly and Overton, is extended to optimization problems over symmetric cones. At each iteration of the Q method, eigenvalues and Jordan frames of decision variables are updated using Newton’s method. We give an
interior point
and a pure Newton’s method based on the Q method. In another paper, the authors have shown that the Q method for secondorder cone programming is accurate. The Q method has also been used to develop a “warmstarting” approach for secondorder cone programming. The machinery of Euclidean Jordan algebra, certain subgroups of the
automorphism group
of symmetric cones, and the
exponential map
is used in the development of the Newton method. Finally we prove that in the presence of certain nondegeneracies the Jacobian of the Newton system is nonsingular at the optimum. Hence the Q method for symmetric cone programming is accurate and can be used to “warmstart” a slightly perturbed symmetric cone program.
Journal:
Journal of Optimization Theory and Applications  J OPTIMIZ THEOR APPL
, vol. 149, no. 1, pp. 102137, 2011
DOI:
10.1007/s109570109777z
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References
(25)
A New PrimalDual InteriorPoint Method for Semidefinite Programming
(
Citations: 28
)
Farid Alizadeh
,
Jeanpierre A. Haeberly
,
Michael L. Overton
Published in 1994.
PrimalDual InteriorPoint Methods for Semidefinite Programming: Convergence Rates, Stability and Numerical Results
(
Citations: 215
)
Farid Alizadeh
,
Jeanpierre A. Haeberly
,
Michael L. Overton
Published in 1994.
The Q method for second order cone programming
(
Citations: 5
)
Yu Xia
,
Farid Alizadeh
Journal:
Computers & Operations Research  CoR
, vol. 35, no. 5, pp. 15101538, 2008
Using SeDuMi 1.02, A Matlab toolbox for optimization over symmetric cones
(
Citations: 1701
)
Jos F. Sturm
Journal:
Optimization Methods & Software  OPTIM METHOD SOFTW
, vol. 11, no. 14, pp. 625653, 1999
A Newton’s method for perturbed secondorder cone programs
(
Citations: 4
)
Yu Xia
Journal:
Computational Optimization and Applications  COMPUT OPTIM APPL
, vol. 37, no. 3, pp. 371408, 2007