Sequential Quadratic Programming Methods Based on Approximating a Projected Hessian Matrix

Abstract

The nonlinear programming problem, namely, minimizing a nonlinear function subject to a set of nonlinear equality and inequality constraints, is considered. Sequential quadratic programming (SQP) methods are particularly effective for solving problems of this nature. It is assumed that first derivatives of the objective and constraint functions are available, but that second derivatives may be too expensive to compute. Instead, the methods typically update a suitable matrix that approximates second derivative information at each iteration. The authors are interested in developing SQP methods that maintain an approximation to second derivative information projected onto the tangent space of the constraints. The main motivation for this work is that only the projected matrix enters into the optimality conditions for the nonlinear problem. Updating projected second derivative information reduces the dimension of the matrix to be recurred; the necessity for introducing an augmenting term that can lead to ill-conditioned matrices is avoided; and one can make use of standard quasi-Newton updates that maintain hereditary positive definiteness. Four possible formulations of the quadratic programming subproblem are discussed, and numerical results indicating that these methods may be useful in practice are presented.