Constrained nonlinear programming

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.