TY - JOUR

T1 - Constrained nonlinear programming

AU - Gill, Philip E.

AU - Murray, Walter

AU - Saunders, Michael A.

AU - Wright, Margaret H.

N1 - Funding Information:
The material contained in this paper is based upon research supported by the Air Force Office of Scientific Research Grant 87-01962; the U.S. Department of Energy Grant DE-FG03-87ER25030; National Science Foundation Grant CCR-8413211; and the Office of Naval Research Contract N00014-87-K-0142.

PY - 1989/1/1

Y1 - 1989/1/1

N2 - This chapter discusses the constrained nonlinear programming. The quadratic programming (QP) problem involves minimizing a quadratic function subject to linear constraints. Quadratic programming is of great interest, and also plays a fundamental role in methods for general nonlinear problems. The basic principle invoked in solving NEP is that of replacing a difficult problem by an easier problem. Penalty functions in their original form are not used, but an understanding of their properties is important for recent methods. Penalty function methods are based on the idea of combining a weighted measure of the constraint violations with the objective function. The chapter discusses the methods based on the optimality conditions for problem NEP. The idea of a quadratic model is a major ingredient in the most successful methods for unconstrained optimization. However, it is shown in the derivation of optimality conditions for NEP that the important curvature is the Lagrangian function. This suggests that quadratic model should be of the Lagrangian function. However, such a model is not the complete representation of the properties of problem NEP. The chapter also discusses the reduced Lagrangian or sequential linearly constrained (SLC) methods. They have been widely used for large-scale optimization problems.

AB - This chapter discusses the constrained nonlinear programming. The quadratic programming (QP) problem involves minimizing a quadratic function subject to linear constraints. Quadratic programming is of great interest, and also plays a fundamental role in methods for general nonlinear problems. The basic principle invoked in solving NEP is that of replacing a difficult problem by an easier problem. Penalty functions in their original form are not used, but an understanding of their properties is important for recent methods. Penalty function methods are based on the idea of combining a weighted measure of the constraint violations with the objective function. The chapter discusses the methods based on the optimality conditions for problem NEP. The idea of a quadratic model is a major ingredient in the most successful methods for unconstrained optimization. However, it is shown in the derivation of optimality conditions for NEP that the important curvature is the Lagrangian function. This suggests that quadratic model should be of the Lagrangian function. However, such a model is not the complete representation of the properties of problem NEP. The chapter also discusses the reduced Lagrangian or sequential linearly constrained (SLC) methods. They have been widely used for large-scale optimization problems.

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U2 - 10.1016/S0927-0507(89)01004-2

DO - 10.1016/S0927-0507(89)01004-2

M3 - Review article

AN - SCOPUS:77957083462

SN - 0927-0507

VL - 1

SP - 171

EP - 210

JO - Handbooks in Operations Research and Management Science

JF - Handbooks in Operations Research and Management Science

IS - C

ER -