where f : lRn → lR is the objective functional, the functions h : lRn → lRm and g : lRn → lRp describe the equality and inequality constraints.

SQP is an iterative procedure which models the NLP for a given iterate x k , k ∈ lN0, by a Quadratic Programming (QP) subproblem, solves that QP subproblem, and then uses the solution to construct a new iterate x k+1. This construction is done in such a way that the sequence (x k )k∈lN0 converges to a local minimum x ∗ of the NLP (4.1a)-(4.1c) as k → ∞. In this sense, the NLP resembles the Newton and quasi-Newton methods for the numerical solution of nonlinear algebraic systems of equations. However, the presence of constraints renders both the analysis and the implementation of SQP methods much more complicated.

The set of points that satisfy the equality and inequality constraints, i.e.,