Mathematical Programming

, Volume 118, Issue 1, pp 1–12 | Cite as

Global convergence of an SQP method without boundedness assumptions on any of the iterative sequences

  • Mikhail V. SolodovEmail author
Short communication


Usual global convergence results for sequential quadratic programming (SQP) algorithms with linesearch rely on some a priori assumptions about the generated sequences, such as boundedness of the primal sequence and/or of the dual sequence and/or of the sequence of values of a penalty function used in the linesearch procedure. Different convergence statements use different combinations of assumptions, but they all assume boundedness of at least one of the sequences mentioned above. In the given context boundedness assumptions are particularly undesirable, because even for non-pathological and well-behaved problems the associated penalty functions (whose descent is used to produce primal iterates) may not be bounded below for any value of the penalty parameter. Consequently, boundedness assumptions on the iterates are not easily justifiable. By introducing a very simple and computationally cheap safeguard in the linesearch procedure, we prove boundedness of the primal sequence in the case when the feasible set is nonempty, convex, and bounded. If, in addition, the Slater condition holds, we obtain a complete global convergence result without any a priori assumptions on the iterative sequences. The safeguard consists of not accepting a further increase of constraints violation at iterates which are infeasible beyond a chosen threshold, which can always be ensured by the proposed modified SQP linesearch criterion.


Sequential quadratic programming Global convergence Nonsmooth penalty function Linesearch Slater condition 

Mathematics Subject Classification (2000)

90C30 65K05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bertsekas D. (1982). Constrained Optimization and Lagrange Multiplier Methods. Academic, New York zbMATHGoogle Scholar
  2. 2.
    Bertsekas D. (1995). Nonlinear Programming. Athena Scientific, Belmont zbMATHGoogle Scholar
  3. 3.
    Bertsekas D. (2003). Convex Analysis and Optimization. Athena Scientific, Belmont zbMATHGoogle Scholar
  4. 4.
    Bonnans J., Gilbert J., Lemaréchal C. and Sagastizábal C. (2003). Numerical Optimization: Theoretical and Practical Aspects. Springer, Berlin zbMATHGoogle Scholar
  5. 5.
    Han S.P. (1976). Superlinearly convergent variable metric algorithms for general nonlinear programming problems. Math. Program. 11: 263–282 CrossRefGoogle Scholar
  6. 6.
    Han S.P. (1977). A globally convergent method for nonlinear programming. J. Optim. Theory Appl. 22: 297–309 zbMATHCrossRefGoogle Scholar
  7. 7.
    Murray W. and Prieto F.J. (1995). A sequential quadratic programming algorithm using an incomplete solution of the subproblem. SIAM J. Optim. 5: 590–640 zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Palomares U.G. and Mangasarian O. (1976). Superlinearly convergent quasi-Newton algorithms for nonlinearly constrained optimization problems. Math. Program. 11: 1–13 zbMATHCrossRefGoogle Scholar
  9. 9.
    Powell M. (1978). The convergence of variable metric methods for nonlinearly constrained optimization calculations. In: Mangasarian, O., Meyer, R. and Robinson, S. (eds) Nonlinear Programming, pp 27–63. Academic, London Google Scholar
  10. 10.
    Robinson S. (1972). A quadratically convergent algorithm for general nonlinear programming problems. Math. Program. 3: 145–156 zbMATHCrossRefGoogle Scholar
  11. 11.
    Robinson S. (1974). Perturbed Kuhn–Tucker points and rates of convergence for a class of nonlinear-programming algorithms. Math. Program. 7: 1–16 zbMATHCrossRefGoogle Scholar
  12. 12.
    Wilson, R.B.: A simplicial method for concave programming. Ph.D. thesis, Graduate School of Business Administration, Harvard University, Cambridge (1963)Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

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

Personalised recommendations