In this paper a family of trust-region interior-point SQP algorithms for the solution of a class of minimization problems with nonlinear equality constraints and simple bounds on some of the variables is described and analyzed. Such nonlinear programs arise e.g. from the discretization of optimal control problems. The algorithms treat states and controls as independent variables. They are designed to take advantage of the structure of the problem. In particular they do not rely on matrix factorizations of the linearized constraints, but use solutions of the linearized state equation and the adjoint equation. They are well suited for large scale problems arising from optimal control problems governed by partial differential equations. The algorithms keep strict feasibility with respect to the bound constraints by using an affine scaling method proposed for a different class of problems by Coleman and Li and they exploit trust--region techniques for equality-constrained optimizatio...

. An algorithm for efficient solution of control constrained optimal control problems is proposed and analyzed. It is based on an active set strategy involving primal as well as dual variables. For discretized problems sufficient conditions for convergence in finitely many iterations are given. Numerical examples are given and the role of strict complementarity condition is discussed. Keywords: Active Set, Augmented Lagrangian, Primal-dual method, Optimal Control. AMS subject classification. 49J20, 49M29 1. Introduction and formulation of the problem. In the recent past significant advances have been made in solving efficiently nonlinear optimal control problems. Most of the proposed methods are based on variations of the sequential quadratic programming (SQP) technique, see for instance [HT, KeS, KuS, K, T] and the references given there. The SQP-algorithm is sequential and each of its iterations requires the solution of a quadratic minimization problem subject to linearized constr...

Tikhonov-Phillips regularization is one of the bestknown regularization methods for inverse problems. A posteriori criteria for determining the regularization parameter ff require solving (A A+ ffI)x = A y ffi () for different values of ff. We investigate two methods for accelerating the standard CG-algorithm for solving the family of systems (). The first one utilizes a stopping criterion for the CG-iterations which depends on ff and ffi. The second method exploits the shifted structure of the linear systems (), which allows to solve () simultaneously for different values of ff. We present numerical experiments for three test problems which illustrate the practical efficiency of the new methods. The experiments as well as theoretical considerations show that run times are accelerated by a factor of at least 3.

This paper investigates the local convergence of the Lagrange-SQP-Newtonmethod applied to an optimal control problem governed by a phase field equation with distributed control. The phase field equation is a system of two semilinear parabolic differential equations. Stability analysis of optimization problems and regularity results for parabolic differential equations are used to proof convergence of the controls with respect to the L 2 (Q) norm and with respect to the L 1 (Q) norm.

. Much progress has been made in constrained nonlinear optimization in the past ten years, but most large-scale problems still represent a considerable obstacle. In this survey paper we will attempt to give an overview of the current approaches, including interior and exterior methods and algorithms based upon trust regions and line searches. In addition, the importance of software, numerical linear algebra and testing will be addressed. We will try to explain why the difficulties arise, how attempts are being made to overcome them and some of the problems that still remain. Although there will be some emphasis on the LANCELOT and CUTE projects, the intention is to give a broad picture of the state-of-the-art. 1 IBM T.J. Watson Research Center, P.O.Box 218, Yorktown Heights, NY 10598, USA 2 Parallel Algorithms Team, CERFACS, 42 Ave. G. Coriolis, 31057 Toulouse Cedex, France 3 Central Computing Department, Rutherford Appleton Laboratory, Chilton, Oxfordshire, OX11 0QX, England ...

: Prior knowledge can be introduced into system identification problems in terms of constraints on the parameter space, or regularizing penalty functions in a prediction error criterion. The contribution of this work is mainly an extention of the well known FPE (Final Prediction Error) statistic to the case when the system identification problem is constrainted and contains a regularization penalty. The FPECR statistic (Final Prediction Error with Constraints and Regularization) is of potential interest as a criterion for selection of both regularization parameters and structural parameters such as order. Keywords: Regularization, Optimization, Parameter Estimation, Nonlinear Systems. 1. INTRODUCTION In practical system identification it is often desirable to introduce prior knowledge into the problem, rather than relying completely on the data. If the model structure is assumed to be fixed, there are still several approaches, cf. Figure 1: (1) Constraints on the parameter space, for e...

. Large scale optimization of systems governed by partial differential equations (PDEs) is a frontier problem in scientific computation. The state-of-the-art for solving such problems is reduced-space quasi-Newton sequential quadratic programming (SQP) methods. These take full advantage of existing PDE solver technology and parallelize well. However, their algorithmic scalability is questionable; for certain problem classes they can be very slow to converge. In this paper we propose a full-space Newton-Krylov SQP method that uses the reducedspace quasi-Newton method as a preconditioner. The new method is fully parallelizable; exploits the structure of and available parallel algorithms for the PDE forward problem; and is quadratically convergent close to a local minimum. We restrict our attention to boundary value problems and we solve a model optimal flow control problem, with both Stokes and Navier-Stokes equations as constraints. Algorithmic comparisons, scalability results, and para...

this paper we propose a preconditioner for the KKT system based on a reduced space quasi-Newton algorithm. Battermann and Heinkenschloss [2] have suggested a preconditioner that is also motivated by reduced methods; the present one can be thought of as a generalization of their method. As in reduced quasi-Newton algorithms, the new preconditioner requires just two linearized flow solves per iteration, but permits the fast convergence associated with full Newton methods. Furthermore, the two flow solves can be approximate, for example using any appropriate flow preconditioner. Finally, the resulting full space SQP parallelizes and scales as well as the flow solver itself. Our method is inspired by the domain-decomposed Schur complement algorithms. In these techniques, reduction onto the interface space requires exact subdomain solves, so one often prefers to iterate within the full space while using a preconditioner based on approximate subdomain solution [8]. Here, decomposition is performed into states and controls, as opposed to subdomain and interface spaces. Below we describe reduced and full space SQP methods and the proposed reduced space-based KKT preconditioner. We also give some performance results on a Cray T3E for a model Stokes flow problem. Our implementation is based on the PETSc library for PDE solution [1], and makes use of PETSc domain-decomposition preconditioners for the approximate flow solves. 2. Reduced SQP methods