We consider an optimal control problem governed by a semi-linear elliptic equation. The controls are distributed. As a consequence, the reduced problems are large. (The size of the reduced problem is approximately half the size of the full problem.)

We use limited BFGS updates to approximate the reduced Hessian of the Lagrangian. The linearized state equation and the adjoint equation which are both elliptic partial differential equations are solved using an iterative method. Which leads to inexact step computations in the SQP method.

The optimization problem has been solved using the TRICE_U algorithms.

More details are given in the following slides. Additional information can be found in the papers
J. E. Dennis, M. Heinkenschloss, and L. N. Vicente: Trust-Region Interior-Point SQP Algorithms for a Class of Nonlinear Programming Problems
M. Heinkenschloss, and L. N. Vicente: Analysis of Inexact Trust-Region Interior-Point SQP Algorithms.