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 <i>et al.</i>, 1983). The problem of concern is assumed to be of the form[EQUATION]where <i>F(x)</i> is a smooth nonlinear function, A_L is a constant matrix, and <i>c(x)</i> is a vector of smooth nonlinear constraint functions. The matrix <i>A_L</i> and the vector <i>c(x)</i> may be empty. Note that <i>upper and lower bounds are specified for all the variables and for all the constraints.</i> This from allows full generality in constraint specification. In particular, the <i>i</i>-th constraint may be defined as an <i>equality</i> by setting <i>l_i</i> = <i>u_i</i>. If certain bounds are not present, the associated elements of <i>l</i> or <i>u</i> can be set to special values that will be treated as - ∞ or +∞.