Journal of Optimization Theory and Applications

, Volume 170, Issue 2, pp 394–418

The Use of Squared Slack Variables in Nonlinear Second-Order Cone Programming

Article

Abstract

In traditional nonlinear programming, the technique of converting a problem with inequality constraints into a problem containing only equality constraints, by the addition of squared slack variables, is well known. Unfortunately, it is considered to be an avoided technique in the optimization community, since the advantages usually do not compensate for the disadvantages, like the increase in the dimension of the problem, the numerical instabilities, and the singularities. However, in the context of nonlinear second-order cone programming, the situation changes, because the reformulated problem with squared slack variables has no longer conic constraints. This fact allows us to solve the problem by using a general-purpose nonlinear programming solver. The objective of this work is to establish the relation between Karush–Kuhn–Tucker points of the original and the reformulated problems by means of the second-order sufficient conditions and regularity conditions. We also present some preliminary numerical experiments.

Keywords

Karush–Kuhn–Tucker conditions Nonlinear second-order cone programming Second-order sufficient condition Slack variables 

Mathematics Subject Classification

90C30 90C46 

References

  1. 1.
    Bertsimas, D., Tsitsiklis, J.N.: Introduction to Linear Optimization, 1st edn. Athena Scientific, Belmont (1997)Google Scholar
  2. 2.
    Dantzig, G.B.: Linear Programming and Extensions. Princeton University Press, Princeton (1963)MATHGoogle Scholar
  3. 3.
    Murtagh, B.A., Saunders, M.A.: Large-scale linearly constrained optimization. Math. Program. 14, 41–72 (1978)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Conn, A.R., Gould, N., Toint, P.L.: A note on exploiting structure when using slack variables. Math. Program. 67, 89–97 (1994)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Spedicato, E.: On a Newton-like method for constrained nonlinear minimization via slack variables. J. Optim. Theory Appl. 36(2), 175–190 (1982)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Tapia, R.A.: A stable approach to Newton’s method for general mathematical programming problems in \(\mathbb{R}^n\). J. Optim. Theory Appl. 14, 453–476 (1974)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Tapia, R.A.: On the role of slack variables in quasi-Newton methods for constrained optimization. In: Dixon, L.C.W., Szegö, G.P. (eds.) Numerical Optimisation of Dynamic Systems, pp. 235–246. North-Holland Publishing Company, Amsterdam (1980)Google Scholar
  8. 8.
    Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)MATHGoogle Scholar
  9. 9.
    Mangasarian, O.L.: Unconstrained Lagrangians in nonlinear programming. SIAM J. Control 13(4), 772–791 (1975)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Robinson, S.M.: Stability theory for systems of inequalities, part II: differentiable nonlinear systems. SIAM J. Numer. Anal. 13(4), 497–513 (1976)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. 95(1), 3–51 (2003)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    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
  13. 13.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  14. 14.
    Fukuda, E.H., Silva, P.J.S., Fukushima, M.: Differentiable exact penalty functions for nonlinear second-order cone programs. SIAM J. Optim. 22(4), 1607–1633 (2012)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kanzow, C., Ferenczi, I., Fukushima, M.: On the local convergence of semismooth Newton methods for linear and nonlinear second-order cone programs without strict complementarity. SIAM J. Optim. 20(1), 297–320 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kato, H., Fukushima, M.: An SQP-type algorithm for nonlinear second-order cone programs. Optim. Lett. 1(2), 129–144 (2007)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Liu, Y.Z., Zhang, L.W.: Convergence of the augmented Lagrangian method for nonlinear optimization problems over second-order cones. J. Optim. Theory Appl. 139(3), 557–575 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Yamashita, H., Yabe, H.: A primal-dual interior point method for nonlinear optimization over second-order cones. Optim. Methods Softw. 24(3), 407–426 (2009)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)CrossRefMATHGoogle Scholar
  20. 20.
    Bonnans, J.F., Ramírez, C.H.: Perturbation analysis of second-order cone programming problems. Math. Program. 104, 205–227 (2005)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Nocedal, J., Wright, S.J.: Numerical Optimization, 1st edn. Springer, New York (1999)CrossRefMATHGoogle Scholar
  22. 22.
    Fourer, R., Gay, D.M., Kernighan, B.W.: A modeling language for mathematical programming. Manag. Sci. 36, 519–554 (1990)CrossRefMATHGoogle Scholar
  23. 23.
    Andreani, R., Birgin, E.G., Martínez, J.M., Schuverdt, M.: On augmented Lagrangian methods with general lower-level constraints. SIAM J. Optim. 18(4), 1286–1309 (2007)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Andreani, R., Birgin, E.G., Martínez, J.M., Schuverdt, M.: Augmented Lagrangian methods under the constant positive linear dependence constraint qualification. Math. Program. 111(1–2), 5–32 (2008)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Physics, Graduate School of InformaticsKyoto UniversityKyotoJapan
  2. 2.Department of Systems and Mathematical Science, Faculty of Science and EngineeringNanzan UniversityNagoyaJapan

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