Some issues in implementing a sequential quadratic programming algorithm

In this note, we consider two of the major issues that have arisen in implementing a sequential quadratic programming (SQP) method for nonlinearly constrained optimization problems (the code NPSOL; Gill et al., 1983). The problem of concern is assumed to be of the form[EQUATION]where F(x) is a smooth nonlinear function, A_L is a constant matrix, and c(x) is a vector of smooth nonlinear constraint functions. The matrix A_L and the vector c(x) may be empty. Note that upper and lower bounds are specified for all the variables and for all the constraints. This from allows full generality in constraint specification. In particular, the i-th constraint may be defined as an equality by setting l_i = u_i. If certain bounds are not present, the associated elements of l or u can be set to special values that will be treated as - ∞ or +∞.