Abstract
Dual interior point methods for solving linear semidefinite programming problems are proposed. These methods are an extension of dual barrier-projection methods for linear programs. It is shown that the proposed methods converge locally at a linear rate provided that the solutions to the primal and dual problems are nondegenerate.
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References
Yu. E. Nesterov and A. S. Nemirovski, Interior Point Polynomial Algorithms in Convex Programming (SIAM, Philadelphia, 1994).
L. Vandenberghe and S. Boyd, “Semidefinite Programming,” SIAM Rev. 38, 49–95 (1996).
M. J. Todd, “Semidefinite optimization,” Acta Numer. 10, 515–560 (2001).
M. Laurent and F. Rendle, “Semidefinite Programming and Integer Programming,” in Discrete Optimization, Handbooks in Operations Research and Management Science (Elsevier, Amsterdam, 2002).
Handbook of Semidefinite Programming, Ed. By H. Wolkowicz, R. Saigal, and L. Vandenberghe (Kluwer, Boston, 2000).
E. De Klerk, Aspects of Semidefinite Programming: Interior Point Algorithms and Selected Applications (Kluwer, Dordrecht, 2004).
F. Alizadeh, “Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization,” SIAM J. Optim. 5, 13–51 (1995).
R. D. C. Monteiro, “Primal-Dual Path-Following Algorithms for Semidefinite Programming,” SIAM J. Optim. 7, 663–678 (1997).
R. D. C. Monteiro, “First- and Second-Order Methods for Semidefinite Programming,” Math. Program., Ser. B 97(1–2), 209–244 (2003).
I. I. Dikin, “Convergence of the Sequence of Dual Variables in the Analysis of a Semidefinite Programming Problem,” Kibern. Sistemn. Anal., No. 2, 156–163 (2003).
B. T. Polyak and P. S. Shcherbakov, “A Randomized Method for Semidefinite Programming Problems,” Stokhastich. Optim. Inform. 2(1), 38–70 (2006).
Yu. G. Evtushenko and V. G. Zhadan, “Dual Barrier-Projection and Barrier-Newton Methods in Linear Programming,” Zh. Vychisl. Mat. Mat. Fiz. 36(7), 30–45 (1996).
M. S. Babynin and V. G. Zhadan, “A Primal Interior Point Method for the Linear Semidefinite Programming Problem,” Comput. Math. Math. Phys. 48, 1746–1767 (2008).
M. Muramatsu, “Affine Scaling Algorithm Fails for Semidefinite Programming,” Math. Program. 83, 393–406 (1998).
J. R. Magnus and H. Neudecker, “The Elimination Matrix: Some Lemmas and Applications,” SIAM J. Alg. Disc. Meth. 1, 422–449 (1980).
J. R. Magnus and H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics (Wiley, New York, 1999; Fizmatlit, Moscow, 2002).
E. E. Tyrtyshnikov, Matrix Analysis and Linear Algebra (Fizmatlit, Moscow, 2007) [in Russian].
V. I. Arnol’d, “On Matrices Depending On Parameters,” Russ. Math. Surveys 26(2), 29–43 (1971).
F. Alizadeh, J.-P. F. Haeberly, and M. L. Overton, “Complementarity and Nondegeneracy in Semidefinite Programming,” Math. Program., Ser. B 7(2), 129–162 (1997).
J. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970; Mir, Moscow, 1975).
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Original Russian Text © V.G. Zhadan, A.A. Orlov, 2011, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2011, Vol. 51, No. 12, pp. 2158–2180.
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Zhadan, V.G., Orlov, A.A. Dual interior point methods for linear semidefinite programming problems. Comput. Math. and Math. Phys. 51, 2031–2051 (2011). https://doi.org/10.1134/S0965542511120189
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DOI: https://doi.org/10.1134/S0965542511120189