Feasible Sequential Quadratic Programming for Finely Discretized Problems from SIP

  • Craig T. Lawrence
  • André L. Tits
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 25)


A Sequential Quadratic Programming algorithm designed to efficiently solve nonlinear optimization problems with many inequality constraints, e.g. problems arising from finely discretized Semi-Infinite Programming, is described and analyzed. The key features of the algorithm are (i) that only a few of the constraints are used in the QP sub-problems at each iteration , and (ii) that every iterate satisfies all constraints.


Search Direction Line Search Accumulation Point Sequential Quadratic Programming Local Convergence 
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  1. [1]
    M. C. Biggs. Constrained minimization using recursive equality quadratic programming. In F. A. Lootsma, editor, Numerical Methods for Non-Linear Optimization, pages 411–428. Academic Press, New York, 1972.Google Scholar
  2. [2]
    P. T. Boggs and J. W. Tolle. Sequential quadratic programming. Acta Numerica, pages 1–51, 1995.Google Scholar
  3. [3]
    I. D. Coope and G. A. Watson. A projected Lagrangian algorithm for semi-infinite programming. Math. Programming, 32:337–356, 1985.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    J. W. Daniel. Stability of the solution of definite quadratic programs. Math. Programming, 5:41–53, 1973.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    C. Gonzaga and E. Polak. On constraint dropping schemes and optimality functions for a class of outer approximation algorithms. SIAM J. Control Optim., 17:477–493, 1979.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    C. Gonzaga, E. Polak, and R. Trahan. An improved algorithm for optimization problems with functional inequality constraints. IEEE Transactions on Automatic Control, AC-25:49–54, 1980.MathSciNetCrossRefGoogle Scholar
  7. [7]
    S. A. Gustafson. A three-phase algorithm for semi-infinite programs. In A. V. Fi-acco and K. O. Kortanek, editors, Semi-Infinite Programming and Applications, Lecture Notes in Control and Information Sciences 215, pages 138–157. Springer Verlag, 1983.CrossRefGoogle Scholar
  8. [8]
    R. Hettich. An implementation of a discretization method for semi-infinite programming. Math. Programming, 34:354–361, 1986.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    R. Hettich and K. O. Kortanek. Semi-infinite programming: theory, methods, and applications. SIAM Rev., 35:380–429, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    R. Hettich and P. Zencke. Numerische Methoden der Approximation und Semiinfiniten Optimierung. Teubner Studienbücher Mathematik, Stuttgart, Germany, 1982.Google Scholar
  11. [11]
    K. C. Kiwiel. Methods of Descent in Nondifferentiable Optimization, Lecture Notes in Mathematics No. 1183. Springer-Verlag, Berlin, 1985.Google Scholar
  12. [12]
    C. T. Lawrence, J. L. Zhou, and A. L. Tits. User’s Guide for CFSQP Version 2.4: A C Code for Solving (Large Scale) Constrained Nonlinear (Minimax) Optimization Problems, Generating Iterates Satisfying All Inequality Constraints, 1996. ISR TR-94–16rl, Institute for Systems Research, University of Maryland (College Park, MD).Google Scholar
  13. [13]
    C. Lemaréchal. Nondifferentiable optimization. In G. Nemhauser, A. Rinooy-Kan, and M. Todd, editors, Optimization, Handbooks in Operations Research and Management Science. Elsevier Science, North Holland, 1989.Google Scholar
  14. [14]
    H. Mine, M. Pukushima, and Y. Tanaka. On the use of ε-most active constraints in an exact penalty function method for nonlinear optimization. IEEE Transactions on Automatic Control, AC-29:1040–1042, 1984.CrossRefGoogle Scholar
  15. [15]
    K. Oettershagen. Ein superlinear konvergenter Algorithmus zur Lösung semiinfiniter Optimierungsprobleme. PhD thesis, Bonn University, 1982.Google Scholar
  16. [16]
    E. R. Panier and A. L. Tits. A globally convergent algorithm with adaptively refined discretization for semi-infinite optimization problems arising in engineering design. IEEE Transactions on Automatic Control, AC-34(8):903–908, 1989.MathSciNetCrossRefGoogle Scholar
  17. [17]
    E. R. Panier and A. L. Tits. On combining feasibility, descent and superlinear convergence in inequality constrained optimization. Math. Programming, 59:261–276, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    E. Polak and L. He. Rate preserving discretization strategies for semi-infinite programming and optimal control. SIAM J. Control and Optimization, 30(3):548–572, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    E. Polak and D. Q. Mayne. An algorithm for optimization problems with functional inequality constraints. IEEE Transactions on Automatic Control, AC-21:184–193, 1976.CrossRefGoogle Scholar
  20. [20]
    E. Polak and A. L. Tits. A recursive quadratic programming algorithm for semi-infinite optimization problems. Appl. Math. Optim., 8:325–349, 1982.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    M. J. D. Powell. A fast algorithm for nonlinearly constrained optimization calculations. In G. A. Watson, editor, Numerical Analysis, Dundee, 1977, Lecture Notes in Mathematics 630, pages 144–157. Springer Verlag, 1978.Google Scholar
  22. [22]
    M. J. D. Powell. A tolerant algorithm for linearly constrained optimization calculations. Math. Programming, 45:547–566, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    R. Reemtsen. Discretization methods for the solution of semi-infinite programming problems. J. Optim. Theory Appl., 71:85–103, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    S. M. Robinson. Perturbed Kuhn-Tucker points and rates of convergence for a class of nonlinear-programming algorithms. Math. Programming, 7:1–16, 1974.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    K. Schittkowski. QLD: A Fortran Code for Quadratic Programming, User’s Guide. Mathematisches Institut, Universität Bayreuth, Germany, 1986.Google Scholar
  26. [26]
    K. Schittkowski. Solving nonlinear programming problems with very many constraints. Optimization, 25:179–196, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    J. L. Zhou and A. L. Tits. An SQP algorithm for finely discretized continuous minimax problems and other minimax problems with many objective functions. SIAM J. on Optimization, pages 461–487, May 1996.Google Scholar
  28. [28]
    G. Zoutendijk. Methods of Feasible Directions. Elsevier Science, Amsterdam, The Netherlands, 1960.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1998

Authors and Affiliations

  • Craig T. Lawrence
    • 1
  • André L. Tits
    • 1
  1. 1.Department of Electrical Engineering and Institute for Systems ResearchUniversity of Maryland, College ParkCollege ParkUSA

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