A New SQP Algorithm for Large-Scale Nonlinear Programming

Abstract

An efficient new SQP algorithm capable of solving large-scale problems is described. It generates descent directions for an $\ell_1$ plus log-barrier merit function and uses a line-search to obtain a sufficient decrease of this function. The unmodified exact Hessian matrix of the Lagrangian function is normally used in the QP subproblem, but this is set to zero if it fails to yield a descent direction for the merit function. The QP problem is solved by an interior-point method using an inexact Newton approach, iterating to an accuracy just sufficient to produce a descent direction in the early stages and tightening the accuracy as we approach a solution. We prove finite termination of the algorithm, at an $\epsilon$-optimal Fritz-John point if feasibility is attained. We also show that if any iterate is close enough to an isolated connected subset of local minimizers, then the iterates converge to this subset. The rate of convergence is Q-quadratic if the subset is an isolated minimizer which satisfies a second-order sufficiency condition, but Q-quadratic convergence to an $\epsilon$-optimal point can still be achieved without any conditions beyond Lipschitz continuity of second-order derivatives. The implementation SQPIPM is designed for problems with many degrees of freedom and is shown to perform well compared with other codes on a range of standard problems.