An adjoint-based SQP algorithm with quasi-Newton Jacobian updates for inequality constrained optimization

We present a sequential quadratic programming (SQP) type algorithm, based on quasi-Newton approximations of Hessian and Jacobian matrices, which is suitable for the solution of general nonlinear programming problems involving equality and inequality constraints. In contrast to most existing SQP methods, no evaluation of the exact constraint Jacobian matrix needs to be performed. Instead, in each SQP iteration only one evaluation of the constraint residuals and two evaluations of the gradient of the Lagrangian function are necessary, the latter of which can efficiently be performed by the reverse mode of automatic differentiation. Factorizations of the Hessian and of the constraint Jacobian are approximated by the recently proposed STR1 update procedure. Inequality constraints are treated by solving within each SQP iteration a quadratic program (QP), the dimension of which equals the number of degrees of freedom. A recently proposed gradient modification in these QPs takes account of Jacobian inexactness in the active set determination. Superlinear convergence of the procedure is shown under mild conditions. The convergence behaviour of the algorithm is analysed using several problems from the Hock-Schittkowski test library. Furthermore, we present numerical results for an optimization problem based on a small periodic adsorption process, where the Jacobian of the equality constraints is dense.