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...

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We present a parallel reduced Hessian SQP method for smooth shape optimization of systems governed by nonlinear boundary value problems, for the case when the number of shape variables is much smaller than the number of state variables. The method avoids nonlinear resolution of the state equations at each design iteration by embedding them as equality constraints in the optimization problem. It makes use of a decomposition into nonorthogonal subspaces that exploits Jacobian and Hessian sparsity in an optimal fashion. The resulting algorithm requires the solution at each iteration of just two linear systems whose coefficients matrices are the state variable Jacobian of the state equations, i.e. the stiffness matrix, and its transpose. The construction and solution of each of these two systems is performed in parallel, as are sensitivity computations associated with the state variables. The conventional parallelism present in a parallel PDE solver---both constructing and solvi...

In this paper we analyze inexact trust-region interior-point (TRIP) sequential quadratic programming (SQP) algorithms for the solution of optimization problems with nonlinear equality constraints and simple bound constraints on some of the variables. Such problems arise in many engineering applications, in particular in optimal control problems with bounds on the control. The nonlinear constraints often come from the discretization of partial differential equations. In such cases the calculation of derivative information and the solution of linearized equations is expensive. Often, the solution of linear systems and derivatives are computed inexactly yielding nonzero residuals. This paper analyzes the effect of the inexactness onto the convergence of TRIP SQP and gives practical rules to control the size of the residuals of these inexact calculations. It is shown that if the size of the residuals is of the order of both the size of the constraints and the trust-region radius, t...

. The focus of this work is on the development of large-scale numerical optimization methods for optimal control of steady incompressible Navier-Stokes flows. The control is affected by the suction or injection of fluid on portions of the boundary, and the objective function represents the rate at which energy is dissipated in the fluid. We develop reduced Hessian sequential quadratic programming methods that avoid converging the flow equations at each iteration. Both quasi-Newton and Newton variants are developed, and compared to the approach of eliminating the flow equations and variables, which is effectively the generalized reduced gradient method. Optimal control problems are solved for two-dimensional flow around a cylinder and three-dimensional flow around a sphere. The examples demonstrate at least an order-of-magnitude reduction in time taken, allowing the optimal solution of flow control problems in as little as half an hour on a desktop workstation. Key words. optimal contr...

We propose a quasi-Newton algorithm for solving large optimization problems with nonlinear equality constraints. It is designed for problems with few degrees of freedom and is motivated by the need to use sparse matrix factorizations. The algorithm incorporates a correction vector that approximates the cross term Z T I/VYp in order to estimate the curvature in both the range and null spaces of the constraints. The algorithm can be considered to be, in some sense, a practical implementation of an algorithm of Coleman and Conn. We give conditions under which local and superlinear convergence is obtained.

The reduced Hessian SQP algorithm presented in [2] is developed in this paper into a practical method for large-scale optimization. The novelty of the algorithm lies in the incorporation of a correction vector that approximates the cross term Z T WYp Y. This improves the stability and robustness of the algorithm without increasing its computational cost. The paper studies how to implement the algorithm e ciently, and presents a set of tests illustrating its numerical performance. An analytic example, showing the bene ts of the correction term, is also presented.