Skip to main content
SpringerLink
Log in

How to project onto extended second order cones

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. Ser. B 95(1), 3–51 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Book  MATH  Google Scholar 

  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)

  7. Gajardo, P., Seeger, A.: Equilibrium problems involving the Lorentz cone. J. Glob. Optim. 58(2), 321–340 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gazi, O.: Understanding Digital Signal Processing, Springer Topics in Signal Processing, vol. 13. Springer, Singapore (2018)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  12. Ko, C.H., Chen, J.S., Yang, C.Y.: Recurrent neural networks for solving second-order cone programs. Neurocomputing 74, 3464–3653 (2011)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Luo, G.M., An, X., Xia, J.Y.: Robust optimization with applications to game theory. Appl. Anal. 88(8), 1183–1195 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  18. Moreau, J.J.: Décomposition orthogonale d’un espace hilbertien selon deux cônes mutuellement polaires. C. R. Acad. Sci. Paris 255, 238–240 (1962)

    MathSciNet  MATH  Google Scholar 

  19. Németh, S., Zhang, G.: Positive operators of extended Lorentz cones. arXiv:1608.07455v2 (2016)

  20. Németh, S.Z., Zhang, G.: Extended Lorentz cones and mixed complementarity problems. J. Glob. Optim. 62(3), 443–457 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Németh, S.Z., Zhang, G.: Extended Lorentz cones and variational inequalities on cylinders. J. Optim. Theory Appl. 168(3), 756–768 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  23. Orlitzky, M., Gowda, M.S.: An improved bound for the Lyapunov rank of a proper cone. Optim. Lett. 10(1), 11–17 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sznajder, R.: The Lyapunov rank of extended second order cones. J. Glob. Optim. 66(3), 585–593 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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 County

  27. Ye, K., Parpas, P., Rustem, B.: Robust portfolio optimization: a conic programming approach. Comput. Optim. Appl. 52(2), 463–481 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IME/UFG, Avenida Esperana, s/n, Campus Samambaia, Goiânia, GO, 74690-900, Brazil

    O. P. Ferreira

  2. School of Mathematics, University of Birmingham, Watson Building, Edgbaston, Birmingham, B15 2TT, UK

    S. Z. Németh

Authors
  1. O. P. Ferreira
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Z. Németh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to O. P. Ferreira.

Additional information

O. P. Ferreira was supported in part by FAPEG and CNPq Grants 305158/2014-7 and 302473/2017-3.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ferreira, O.P., Németh, S.Z. How to project onto extended second order cones. J Glob Optim 70, 707–718 (2018). https://doi.org/10.1007/s10898-017-0587-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-017-0587-9

Keywords

Search

Navigation