Constraint incorporation in optimization Authors Adam B. Levy Department of Mathematics Bowdoin College Full Lenght Paper

First Online: 29 July 2006 Received: 09 April 2005 Accepted: 08 June 2006 DOI :
10.1007/s10107-006-0016-1

Cite this article as: Levy, A.B. Math. Program. (2007) 110: 615. doi:10.1007/s10107-006-0016-1
Abstract
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.

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

References 1.

Ben-Tal A., Zibulevsky M. (1997) Penalty/barrier multiplier methods for convex programming problems. SIAM J. Optim. 7, 347–366

MATH CrossRef MathSciNet 2.

Boggs P., Tolle J. (1995) Sequential quadratic programming Acta numerica, Acta Numer. Cambridge University Press, Cambridge, pp. 1–51

3.

Boukari D., Fiacco A.V. (1995) Survey of penalty, exact-penalty and multiplier methods from 1968 to 1993. Optimization 32, 301–334

MATH MathSciNet 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, 27–48

MATH MathSciNet 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, 153–161

MATH CrossRef MathSciNet 6.

Demyanov V.F., DiPillo G., Facchinei F. (1998) Exact penalization via Dini and Hadamard conditional derivatives. Optim. Methods Softw. 9, 19–36

MATH MathSciNet 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 , 209–253

8.

DiPillo G., Facchinei F. (1995) Exact barrier function methods for Lipschitz programs. Appl. Math. Optim. 32, 1–31

CrossRef MathSciNet 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)

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, 49–68

CrossRef MathSciNet 11.

DiPillo G., Grippo L., Lucidi S. (1997) Smooth transformation of the generalized minimax problem. J. Optim. Theory Appl. 95, 1–24

CrossRef MathSciNet 12.

DiPillo G., Grippo L., Lucidi S. (1993) A smooth method for the finite minimax problem. Math. Program. 60, 187–214

CrossRef MathSciNet 13.

DiPillo G., Lucidi S., Palagi L. (1999) A shifted-barrier primal-dual algorithm model for linearly constrained optimization problems. Comput. Optim. Appl. 12, 157–188

CrossRef MathSciNet 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, 129–148

MathSciNet 15.

Facchinei F. (1997) Robust recursive quadratic programming algorithm model with global and superlinear convergence properties. J. Optim. Theory Appl. 92, 543–579

MATH CrossRef MathSciNet 16.

Facchinei F. (1991) Exact penalty functions and Lagrange multipliers. Optimization 22, 579–606

MATH MathSciNet 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, 85–122

MATH CrossRef MathSciNet 18.

Facchinei F., Lucidi S. (1998) Convergence to second-order stationary points in inequality constrained optimization. Math. Oper. Res. 23, 746–766

MATH MathSciNet CrossRef 19.

Facchinei F., Lucidi S. (1992) A class of penalty functions for optimization problems with bound constraints. Optimization 26, 239–259

MATH MathSciNet 20.

Fletcher R. (1987) Practical Methods of Optimization. Wiley, New York

MATH 21.

Nocedal J., Wright S.J. (1999) Numerical Optimization. Springer, Berlin Heidelberg New York

MATH CrossRef 22.

Rockafellar R.T., Wets R.J.-B. (1998) Variational Analysis. Springer, Berlin Heidelberg New York

MATH 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)