V.G. Zhadan. A variant of the affine-scaling method for a cone programming problem on a second-order cone ... P. 114-124.

A linear cone programming problem in which the cone is the direct product of second-order cones (Lorentz cones) is considered. For its solution we propose a direct affine-scaling type method generalizing the corresponding method used in linear programming. The method can be considered as a special way to solve a system of necessary and sufficient optimality conditions for a pair of mutually dual cone programming problems. These conditions are used to derive the dependence of the dual variables on the primal variables, and the dependence is substituted into the complementarity condition. The obtained system of equations is solved with respect to the primal variables by the simple iteration method. The starting points in the method belong to the cone but do not necessarily satisfy the linear equality-type constraints. The local linear convergence of the method is proved under the assumption that the solutions of the primal and dual problems are nondegenerate and strictly complementary.  

Keywords: cone programming, second-order cone, affine-scaling method, local convergence.  

The paper was received by the Editorial Office on May 31, 2017.

Vitalii Grigor'evich Zhadan, Dr. Phys.-Math. Sci., Prof., Dorodnicyn Computing Centre, FRC CSC RAS,  Moscow, 119333 Russia, e-mail: zhadan@ccas.ru .

REFERENCES

1.   Alizadeh F., Goldfarb D. Second-order cone programming. Math. Program., Ser. B, 2003, vol. 95, pp. 3–51. doi: 10.1016/j.ifacol.2015.11.079.

2.   Lobo M.S., Vandenberghe L., Boyd S., Lebret H.. Applications of second order cone programming. Linear Algebra Appl., 1998, vol. 284, pp. 193–228. doi: 10.1016/S0024-3795(98)10032-0 .

3.   Anjos M. F., Lasserre J. B., eds. Handbook on semidefinite, cone and polynomial optimization: theory, algorithms, software and applications. New York: Springer, 2011, 960 p. doi: 10.1007/978-1-4614-0769-0 .

4.   Nesterov Y. E., Todd M. J. Primal-dual interior-point methods for self-scaled cones. SIAM J. Optim., 1998, vol. 8, no. 2, pp. 324–364. doi: 10.1137/S1052623495290209 .

5.   Monteiro R. D. C., Tsuchia T. Polynomial convergence of primal-dual algorithms for the second-order cone program bazed on MZ-family directions. Math. Program., 2000, vol. 88, pp. 61–83. doi: 10.1007/s101070000137 .

6.   Muramatsu M. A pivoting procedure for a class of second-order cone programming. Optim. Methods Softw, 2006, vol. 21, no. 2, pp. 295–315. doi: 10.1080/10556780500094697 .

7.   Evtushenko Yu., Zhadan V. Stable barrier-projection and barrier-newton methods in linear programming. Comput. Optimiz. Appl., 1994, vol. 3, no. 4, pp. 289–304.

8.   Babynin M.S., Zhadan V.G. A primal interior point method for linear semidefinite programming problem. Comput. Math. and Math. Phys., 2008, vol. 48, no. 10, pp. 1746-1767. doi: 10.1134/S0965542508100035 .

9.   Pataki G. Cone-LP’s and semidefinite programs: geometry and simplex-type method. Proc. Conf. on Integer Programming and Combinatorial Optimization (IPCO 5), 1996, pp. 1–13.

10.   Ortega J.M., Rheinboldt W.C. Iterative solution of nonlinear equations in several variables. New York; London: Academic Press, 1970, 592 p. ISBN: 978-0-12-528550-6 . Translated under the title “Iteratsionnye metody resheniya sistem nelineinykh uravnenii so mnogimi neizvestnymi”, Moscow, Mir, 1975, 560 p.