Computational Optimization and Applications

, Volume 39, Issue 2, pp 143–160 | Cite as

On local convergence of sequential quadratically-constrained quadratic-programming type methods, with an extension to variational problems



We consider the class of quadratically-constrained quadratic-programming methods in the framework extended from optimization to more general variational problems. Previously, in the optimization case, Anitescu (SIAM J. Optim. 12, 949–978, 2002) showed superlinear convergence of the primal sequence under the Mangasarian-Fromovitz constraint qualification and the quadratic growth condition. Quadratic convergence of the primal-dual sequence was established by Fukushima, Luo and Tseng (SIAM J. Optim. 13, 1098–1119, 2003) under the assumption of convexity, the Slater constraint qualification, and a strong second-order sufficient condition. We obtain a new local convergence result, which complements the above (it is neither stronger nor weaker): we prove primal-dual quadratic convergence under the linear independence constraint qualification, strict complementarity, and a second-order sufficiency condition. Additionally, our results apply to variational problems beyond the optimization case. Finally, we provide a necessary and sufficient condition for superlinear convergence of the primal sequence under a Dennis-Moré type condition.


Quadratically constrained quadratic programming Karush-Kuhn-Tucker system Variational inequality Quadratic convergence Superlinear convergence Dennis-Moré condition 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anitescu, M.: A superlinearly convergent sequential quadratically constrained quadratic programming algorithm for degenerate nonlinear programming. SIAM J. Optim. 12, 949–978 (2002) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Boggs, B.T., Tolle, J.W.: Sequential quadratic programming. Acta Numer. 4, 1–51 (1996) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000) MATHGoogle Scholar
  4. 4.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1990) MATHGoogle Scholar
  5. 5.
    Daryina, A.N., Izmailov, A.F., Solodov, M.V.: A class of active-set Newton methods for mixed complementarity problems. SIAM J. Optim. 15, 409–429 (2004) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dennis, J.E., Moré, J.J.: A characterization of superlinear convergence and its application to quasi-Newton methods. Math. Comput. 28, 549–560 (1974) MATHCrossRefGoogle Scholar
  7. 7.
    Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003) Google Scholar
  8. 8.
    Fukushima, M., Luo, Z.-Q., Tseng, P.: A sequential quadratically constrained quadratic programming method for differentiable convex minimization. SIAM J. Optim. 13, 1098–1119 (2003) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Izmailov, A.F., Solodov, M.V.: Karush-Kuhn-Tucker systems: regularity conditions, error bounds and a class of Newton-type methods. Math. Program. 95, 631–650 (2003) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Josephy, N.H.: Newton’s method for generalized equations. Technical Summary Report 1965, Mathematics Research Center, University of Wisconsin, Madison, Wisconsin (1979) Google Scholar
  11. 11.
    Kruk, S., Wolkowicz, H.: Sequential, quadratically constrained, quadratic programming for general nonlinear programming. In: Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds.) Handbook of Semidefinite Programming, pp. 563–575. Kluwer Academic, Dordrecht (2000) Google Scholar
  12. 12.
    Lobo, M.S., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second-order cone programming. Linear Algebra Appl. 284, 193–228 (1998) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Maratos, N.: Exact penalty function algorithms for finite dimensional and control optimization problems. Ph.D. thesis, Imperial College, University of London (1978) Google Scholar
  14. 14.
    Monteiro, R.D.C., Tsuchiya, T.: Polynomial convergence of primal-dual algorithms for the second-order cone programs based on the MZ-family of directions. Math. Program. 88, 61–83 (2000) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Nesterov, Y.E., Nemirovskii, A.S.: Interior Point Polynomial Methods in Convex Programming: Theory and Applications. SIAM, Philadelphia (1993) Google Scholar
  16. 16.
    Panin, V.M.: A second-order method for discrete min-max problem. USSR Comput. Math. Math. Phys. 19, 90–100 (1979) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Panin, V.M.: Some methods of solving convex programming problems. USSR Comput. Math. Math. Phys. 21, 57–72 (1981) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Powell, M.J.D.: Variable metric methods for constrained optimization. In: Bachem, A., Grötschel, M., Korte, B. (eds.) Mathematical Programming: The State of Art, pp. 288–311. Springer, Berlin (1983) Google Scholar
  19. 19.
    Solodov, M.V.: On the sequential quadratically constrained quadratic programming methods. Math. Oper. Res. 29, 64–79 (2004) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Tsuchiya, T.: A convergence analysis of the scale-invariant primal-dual path-following algorithms for second-order cone programming. Optim. Methods Softw. 11, 141–182 (1999) CrossRefMathSciNetGoogle Scholar
  21. 21.
    Wiest, E.J., Polak, E.: A generalized quadratic programming-based phase-I–phase-II method for inequality-constrained optimization. Appl. Math. Optim. 26, 223–252 (1992) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Instituto de Matemática Pura e AplicadaRio de JaneiroBrazil

Personalised recommendations