Mathematical Programming

, Volume 110, Issue 3, pp 615–639 | Cite as

Constraint incorporation in optimization

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Numerical methods for solving constrained optimization problems need to incorporate the constraints in a manner that satisfies essentially competing interests; the incorporation needs to be simple enough that the solution method is tractable, yet complex enough to ensure the validity of the ultimate solution. We introduce a framework for constraint incorporation that identifies a minimal acceptable level of complexity and defines two basic types of constraint incorporation which (with combinations) cover nearly all popular numerical methods for constrained optimization, including trust region methods, penalty methods, barrier methods, penalty-multiplier methods, and sequential quadratic programming methods. The broad application of our framework relies on addition and chain rules for constraint incorporation which we develop here.


Constrained optimization Numerical optimization Penalty methods Barrier methods Trust region methods Penalty-multiplier methods Sequential quadratic programming Variational analysis 


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  1. 1.
    Ben-Tal A., Zibulevsky M. (1997) Penalty/barrier multiplier methods for convex programming problems. SIAM J. Optim. 7, 347–366MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Boggs P., Tolle J. (1995) Sequential quadratic programming Acta numerica, Acta Numer. Cambridge University Press, Cambridge, pp. 1–51Google Scholar
  3. 3.
    Boukari D., Fiacco A.V. (1995) Survey of penalty, exact-penalty and multiplier methods from 1968 to 1993. Optimization 32, 301–334MATHMathSciNetGoogle Scholar
  4. 4.
    Byrd R., Gould N., Nocedal J., Waltz R. (2004) An active set algorithm for nonlinear programming using linear programming and equality constrained subproblems. Math. Program. 100, 27–48MATHMathSciNetGoogle Scholar
  5. 5.
    Contaldi G., DiPillo G., Lucidi S. (1993) A continuously differentiable exact penalty function for nonlinear programming problems with unbounded feasible set. Oper. Res. Lett. 14, 153–161MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Demyanov V.F., DiPillo G., Facchinei F. (1998) Exact penalization via Dini and Hadamard conditional derivatives. Optim. Methods Softw. 9, 19–36MATHMathSciNetGoogle Scholar
  7. 7.
    DiPillo G. (1994) Exact Penalty Methods. Algorithms for Continuous Optimization (Il Ciocco, 1993). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer, Dordrecht 434, 209–253Google Scholar
  8. 8.
    DiPillo G., Facchinei F. (1995) Exact barrier function methods for Lipschitz programs. Appl. Math. Optim. 32, 1–31CrossRefMathSciNetGoogle Scholar
  9. 9.
    DiPillo G., Facchinei F. Regularity conditions and exact penalty functions in Lipschitz programming problems. Nonsmooth optimization: methods and applications (Erice, 1991) pp. 107–120. Gordon and Breach, Montreux, (1992)Google Scholar
  10. 10.
    DiPillo G., Facchinei F., Grippo L. (1992) An RQP algorithm using a differentiable exact penalty function for inequality constrained problems. Math. Program. Series A 55, 49–68CrossRefMathSciNetGoogle Scholar
  11. 11.
    DiPillo G., Grippo L., Lucidi S. (1997) Smooth transformation of the generalized minimax problem. J. Optim. Theory Appl. 95, 1–24CrossRefMathSciNetGoogle Scholar
  12. 12.
    DiPillo G., Grippo L., Lucidi S. (1993) A smooth method for the finite minimax problem. Math. Program. 60, 187–214CrossRefMathSciNetGoogle Scholar
  13. 13.
    DiPillo G., Lucidi S., Palagi L. (1999) A shifted-barrier primal-dual algorithm model for linearly constrained optimization problems. Comput. Optim. Appl. 12, 157–188CrossRefMathSciNetGoogle Scholar
  14. 14.
    DiPillo G., Lucidi S., Palagi L. (1993) An exact penalty-Lagrangian approach for a class of constrained optimization problems with bounded variables. Optimization 28, 129–148MathSciNetGoogle Scholar
  15. 15.
    Facchinei F. (1997) Robust recursive quadratic programming algorithm model with global and superlinear convergence properties. J. Optim. Theory Appl. 92, 543–579MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Facchinei F. (1991) Exact penalty functions and Lagrange multipliers. Optimization 22, 579–606MATHMathSciNetGoogle Scholar
  17. 17.
    Facchinei F., Liuzzi G., Lucidi S. (2003) A truncated Newton method for the solution of large-scale inequality constrained minimization problems. Comput. Optim. Appl. 25, 85–122MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Facchinei F., Lucidi S. (1998) Convergence to second-order stationary points in inequality constrained optimization. Math. Oper. Res. 23, 746–766MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Facchinei F., Lucidi S. (1992) A class of penalty functions for optimization problems with bound constraints. Optimization 26, 239–259MATHMathSciNetGoogle Scholar
  20. 20.
    Fletcher R. (1987) Practical Methods of Optimization. Wiley, New YorkMATHGoogle Scholar
  21. 21.
    Nocedal J., Wright S.J. (1999) Numerical Optimization. Springer, Berlin Heidelberg New YorkMATHCrossRefGoogle Scholar
  22. 22.
    Rockafellar R.T., Wets R.J.-B. (1998) Variational Analysis. Springer, Berlin Heidelberg New YorkMATHGoogle Scholar
  23. 23.
    Sadjadi S., Ponnambalam K.: Advances in trust region algorithms for constrained optimization. In: Proceedings of the Stieltjes Workshop on High Performance Optimization Techniques (Delft), Appl. Numer. Math. 29, 423–443 (1999)Google Scholar

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© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of MathematicsBowdoin CollegeBrunswickUSA

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