Dual Newton’s Methods for Linear Second-Order Cone Programming

  • Vitaly ZhadanEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12095)


The linear second-order cone programming problem is considered. For its solution, two dual Newton’s methods are proposed. These methods are constructed with the help of optimality conditions. The nonlinear system of equations, obtained from the optimality conditions and depended only from dual variables, is solved by the Newton method. Under the assumption that there exist strictly complementary solutions of both primal and dual problems the local convergence of the methods with super-linear rate is proved.


Linear second-order cone programming Dual Newton’s method Local convergence Super-linear rate of convergence 


  1. 1.
    Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. Ser. B. 95, 3–51 (2003)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Lobo, M.S., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second order cone programming. Linear Algebra Appl. 284, 193–228 (1998)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Anjos, M.F., Lasserre, J.B. (eds.): Handbook of Semidefinite, Cone and Polynomial Optimization: Theory, Algorithms, Software and Applications, p. 915. Springer, New York (2011). Scholar
  4. 4.
    Nesterov, Y.E., Todd, M.J.: Primal-dual interior-point methods for self-scaled cones. SIAM. J. Optim. 8, 324–364 (1998)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Monteiro, R.D.C., Tsuchiya, T.: Polynomial convergence of primal-dual algorithms for second-order cone program based on the MZ-family of directions. Math. Program. 88(1), 61–83 (2000)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Zhadan, V.: Dual multiplicative-barrier methods for linear second-order cone programming. In: Jaćimović, M., Khachay, M., Malkova, V., Posypkin, M. (eds.) OPTIMA 2019. CCIS, vol. 1145, pp. 295–310. Springer, Cham (2020). Scholar
  7. 7.
    Evtushenko, Y.G., Zhadan, V.G.: Dual barrier-projection and barrier-newton methods for linear programming. Comp. Maths. Math. Phys. 36(7), 847–859 (1996)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Zhadan, V.G.: Primal Newton method for the linear cone programming problem. Comput. Mathe. Mathe. Physics. 58(2), 207–214 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Dorodnicyn Computing Centre, FRC “Computer Science and Control” of RASMoscowRussia
  2. 2.Moscow Institute of Physics and Technology (State Research University)DolgoprudnyRussia

Personalised recommendations