Journal of Global Optimization

, Volume 70, Issue 4, pp 707–718 | Cite as

How to project onto extended second order cones

  • O. P. Ferreira
  • S. Z. Németh


The extended second order cones were introduced by Németh and Zhang (J Optim Theory Appl 168(3):756–768, 2016) for solving mixed complementarity problems and variational inequalities on cylinders. Sznajder (J Glob Optim 66(3):585–593, 2016) determined the automorphism groups and the Lyapunov or bilinearity ranks of these cones. Németh and Zhang (Positive operators of extended Lorentz cones, 2016. arXiv:1608.07455v2) found both necessary conditions and sufficient conditions for a linear operator to be a positive operator of an extended second order cone. In this note we give formulas for projecting onto the extended second order cones. In the most general case the formula depends on a piecewise linear equation for one real variable which is solved by using numerical methods.


Semi-smooth equation Extended second order cone Metric projection Piecewise linear Newton method 


  1. 1.
    Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. Ser. B 95(1), 3–51 (2003)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Barrios, J.G., Bello Cruz, J.Y., Ferreira, O.P., Németh, S.Z.: A semi-smooth Newton method for a special piecewise linear system with application to positively constrained convex quadratic programming. J. Comput. Appl. Math. 301, 91–100 (2016)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bello Cruz, J.Y., Ferreira, O.P., Németh, S., Prudente, L.F.: A semi-smooth Newton method for projection equations and linear complementarity problems with respect to the second order cone. Linear Algebra Appl. 513, 160–181 (2017)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chen, J.S., Tseng, P.: An unconstrained smooth minimization reformulation of the second-order cone complementarity problem. Math. Program. Ser. B 104(2–3), 293–327 (2005)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chi, C.Y., Li, W.C., Lin, C.H.: Convex Optimization for Signal Processing and Communications: From Fundamentals to Applications. CRC Press, Boca Raton (2017)CrossRefMATHGoogle Scholar
  6. 6.
    Fukushima, M., Luo, Z.Q., Tseng, P.: Smoothing functions for second-order-cone complementarity problems. SIAM J. Optim. 12(2), 436–460 (2001/02)Google Scholar
  7. 7.
    Gajardo, P., Seeger, A.: Equilibrium problems involving the Lorentz cone. J. Glob. Optim. 58(2), 321–340 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gazi, O.: Understanding Digital Signal Processing, Springer Topics in Signal Processing, vol. 13. Springer, Singapore (2018)CrossRefMATHGoogle Scholar
  9. 9.
    Gowda, M.S., Tao, J.: On the bilinearity rank of a proper cone and Lyapunov-like transformations. Math. Program. Ser. A 147(1–2), 155–170 (2014)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gowda, M.S., Trott, D.: On the irreducibility, Lyapunov rank, and automorphisms of special Bishop–Phelps cones. J. Math. Anal. Appl. 419(1), 172–184 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. I. Fundamentals, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 305. Springer, Berlin (1993)MATHGoogle Scholar
  12. 12.
    Ko, C.H., Chen, J.S., Yang, C.Y.: Recurrent neural networks for solving second-order cone programs. Neurocomputing 74, 3464–3653 (2011)CrossRefGoogle Scholar
  13. 13.
    Kong, L., Xiu, N., Han, J.: The solution set structure of monotone linear complementarity problems over second-order cone. Oper. Res. Lett. 36(1), 71–76 (2008)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lobo, M.S., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second-order cone programming. Linear Algebra Appl. 284(1–3), 193–228 (1998)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Luo, G.M., An, X., Xia, J.Y.: Robust optimization with applications to game theory. Appl. Anal. 88(8), 1183–1195 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Malik, M., Mohan, S.R.: On Q and R\(_0\) properties of a quadratic representation in linear complementarity problems over the second-order cone. Linear Algebra Appl. 397, 85–97 (2005)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Markowitz, H.M.: Portfolio Selection: Efficient Diversification of Investments. Cowles Foundation for Research in Economics at Yale University, Monograph 16. Wiley, New York; Chapman & Hall, London (1959)Google Scholar
  18. 18.
    Moreau, J.J.: Décomposition orthogonale d’un espace hilbertien selon deux cônes mutuellement polaires. C. R. Acad. Sci. Paris 255, 238–240 (1962)MathSciNetMATHGoogle Scholar
  19. 19.
    Németh, S., Zhang, G.: Positive operators of extended Lorentz cones. arXiv:1608.07455v2 (2016)
  20. 20.
    Németh, S.Z., Zhang, G.: Extended Lorentz cones and mixed complementarity problems. J. Glob. Optim. 62(3), 443–457 (2015)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Németh, S.Z., Zhang, G.: Extended Lorentz cones and variational inequalities on cylinders. J. Optim. Theory Appl. 168(3), 756–768 (2016)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Nishimura, R., Hayashi, S., Fukushima, M.: Robust Nash equilibria in \(N\)-person non-cooperative games: uniqueness and reformulation. Pac. J. Optim. 5(2), 237–259 (2009)MathSciNetMATHGoogle Scholar
  23. 23.
    Orlitzky, M., Gowda, M.S.: An improved bound for the Lyapunov rank of a proper cone. Optim. Lett. 10(1), 11–17 (2016)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Rudolf, G., Noyan, N., Papp, D., Alizadeh, F.: Bilinear optimality constraints for the cone of positive polynomials. Math. Program. Ser. B 129(1), 5–31 (2011)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Sznajder, R.: The Lyapunov rank of extended second order cones. J. Glob. Optim. 66(3), 585–593 (2016)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Trott, D.W.: Topheavy and special Bishop–Phelps cones, Lyapunov rank, and related topics. ProQuest LLC, Ann Arbor (2014). Thesis (Ph.D.), University of Maryland, Baltimore CountyGoogle Scholar
  27. 27.
    Ye, K., Parpas, P., Rustem, B.: Robust portfolio optimization: a conic programming approach. Comput. Optim. Appl. 52(2), 463–481 (2012)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Yonekura, K., Kanno, Y.: Second-order cone programming with warm start for elastoplastic analysis with von Mises yield criterion. Optim. Eng. 13(2), 181–218 (2012)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Zhang, L.L., Li, J.Y., Zhang, H.W., Pan, S.H.: A second order cone complementarity approach for the numerical solution of elastoplasticity problems. Comput. Mech. 51(1), 1–18 (2013)MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.IME/UFGGoiâniaBrazil
  2. 2.School of MathematicsUniversity of BirminghamBirminghamUK

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