Skip to main content
SpringerLink
Log in

A Newton’s method for perturbed second-order cone programs

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

We develop optimality conditions for the second-order cone program. Our optimality conditions are well-defined and smooth everywhere. We then reformulate the optimality conditions into several systems of equations. Starting from a solution to the original problem, the sequence generated by Newton’s method converges Q-quadratically to a solution of the perturbed problem under some assumptions. We globalize the algorithm by (1) extending the gradient descent method for differentiable optimization to minimizing continuous functions that are almost everywhere differentiable; (2) finding a directional derivative of the equations. Numerical examples confirm that our algorithm is good for “warm starting” second-order cone programs—in some cases, the solution of a perturbed instance is hit in two iterations. In the progress of our algorithm development, we also generalize the nonlinear complementarity function approach for two variables to several variables.

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. Adler, I., Alizadeh, F.: Primal–dual interior point algorithms for convex quadratically constrained and semidefinite optimization problems. Technical report RRR 46-95, RUTCOR, Rutgers University (1995)

  2. Alizadeh, F.: Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM J. Optim. 5(1), 13–51 (1995)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. Alizadeh, F., Schmieta, S.H.: Optimization with semidefinite, quadratic and linear constraints. Technical report RRR 23-97, RUTCOR, Rutgers University (1997)

  5. Bach, F.R., Lanckriet, G.R.G., Jordan, M.I.: Multiple kernel learning, conic duality, and the SMO algorithm. In: ICML ’04: Proceedings of the Twenty-first International Conference on Machine Learning, p. 6. ACM, New York (2004)

    Chapter  Google Scholar 

  6. Benson, H.Y., Vanderbei, R.J.: Solving problems with semidefinite and related constraints using interior-point methods for nonlinear programming. Math. Program. Ser. B 95(2), 279–302 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific (1999)

  8. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley Interscience, New York (1983)

    MATH  Google Scholar 

  9. De Luca, T., Facchinei, F., Kanzow, C.: A theoretical and numerical comparison of some semismooth algorithms for complementarity problems. Comput. Optim. Appl. 16(2), 173–205 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dennis Jr., J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs (1983)

    MATH  Google Scholar 

  11. Evtushenko, Yu.G., Purtov, V.A.: Sufficient conditions for a minimum for nonlinear programming problems. Dokl. Akad. Nauk SSSR 278(1), 24–27 (1984)

    MathSciNet  Google Scholar 

  12. Faraut, J., Korányi, A.: Analysis on Symmetric Cones. Clarendon/Oxford University Press, New York (1994)

    MATH  Google Scholar 

  13. Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39(4), 669–713 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  14. Fischer, A.: A special Newton-type optimization method. Optimization 24(3–4), 269–284 (1992)

    MATH  MathSciNet  Google Scholar 

  15. Fischer, A., Jiang, H.: Merit functions for complementarity and related problems: a survey. Comput. Optim. Appl. 17(2–3), 159–182 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Fukushima, M., Luo, Z.-Q., Tseng, P.: Smoothing functions for second-order-cone complementarity problems. SIAM J. Optim. 12(2), 436–460 (2001) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  17. Garey, M.R., Graham, R.L., Johnson, D.S.: The complexity of computing Steiner minimal trees. SIAM J. Appl. Math. 32(4), 835–859 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  18. Gondzio, J., Grothey, A.: Reoptimization with the primal–dual interior point method. SIAM J. Optim. 13(3), 842–864 (2002) (electronic, 2003)

    Article  MathSciNet  Google Scholar 

  19. Grippo, L., Lampariello, F., Lucidi, S.: A nonmonotone line search technique for Newton’s method. SIAM J. Numer. Anal. 23(4), 707–716 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  20. Lobo, M.S., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second-order cone programming. Linear Algebra Appl. 284(1–3), 193–228 (1998). Also in: ILAS Symposium on Fast Algorithms for Control, Signals and Image Processing, Winnipeg, MB, 1997

    Article  MATH  MathSciNet  Google Scholar 

  21. Lustig, I.J., Marsten, R.E., Shanno, D.F.: Computational experience with a globally convergent primal–dual predictor–corrector algorithm for linear programming. Math. Program. Ser. A 66(1), 123–135 (1994)

    Article  MathSciNet  Google Scholar 

  22. Mangasarian, O.L.: Equivalence of the complementarity problem to a system of nonlinear equations. SIAM J. Appl. Math. 31(1), 89–92 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15(6), 959–972 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  24. Mitchell, J.E., Todd, M.J.: Solving combinatorial optimization problems using Karmarkar’s algorithm. Math. Program. Ser. A 56(3), 245–284 (1992)

    Article  MathSciNet  Google Scholar 

  25. Pang, J.-S., Qi, L.: A globally convergent Newton method for convex sc1 minimization problems. J. Optim. Theory Appl. 85(3), 633–648 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  26. Pang, J.-S.: Newton’s method for B-differentiable equations. Math. Oper. Res. 15(2), 311–341 (1990)

    MATH  MathSciNet  Google Scholar 

  27. Polak, E., Mayne, D.Q.: Algorithm models for nondifferentiable optimization. SIAM J. Control Optim. 23(3), 477–491 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  28. Qi, L.Q.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18(1), 227–244 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  29. Qi, L.Q., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. Ser. A 58(3), 353–367 (1993)

    Article  MathSciNet  Google Scholar 

  30. Qi, L., Jiang, H.: Semismooth Karush–Kuhn–Tucker equations and convergence analysis of Newton and quasi-Newton methods for solving these equations. Math. Oper. Res. 22(2), 301–325 (1997)

    MATH  MathSciNet  Google Scholar 

  31. Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, vol. 28. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  32. Wolfe, P.: A method of conjugate subgradients for minimizing nondifferentiable functions. Math. Program. Stud. 3, 145–173 (1975)

    MathSciNet  Google Scholar 

  33. Wright, S.J.: Primal–Dual Interior-Point Methods. SIAM, Philadelphia (1997)

    MATH  Google Scholar 

  34. Xia, Y., Alizadeh, F.: The Q method for second-order cone programming. Comput. Oper. Res. (2006). doi:10.1016/j.cor.2006.08.009 (available on-line November 2006)

  35. Xue, G., Ye, Y.: An efficient algorithm for minimizing a sum of Euclidean norms with applications. SIAM J. Optim. 7(4), 1017–1036 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  36. Yildirim, E.A., Wright, S.J.: Warm-start strategies in interior-point methods for linear programming. SIAM J. Optim. 12(3), 782–810 (2002)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Operations Research and Econometrics (CORE), Université Catholique de Louvain, 34 Voie du Roman Pays, B-1348, Louvain-la-Neuve, Belgium

    Yu Xia

Authors
  1. Yu Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu Xia.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xia, Y. A Newton’s method for perturbed second-order cone programs. Comput Optim Appl 37, 371–408 (2007). https://doi.org/10.1007/s10589-007-9023-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-007-9023-2

Keywords

Search

Navigation