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...

We consider a distributed optimal control problem governed by a semilinear parabolic equation, where constraints on the control and on the state are given. Aiming to show the existence of regular Lagrange multipliers we follow a linearization approach together with a two-norm technique. The theory is applied to derive a generalized bang-bang principle. 1 This work was supported by EEC, HCM Contract CHRX-CT94-0471 1. INTRODUCTION In this paper we investigate some optimal control problems where the state equation is a semilinear parabolic equation. In addition, we consider constraints on both the control and the state. Our main purpose is to get some Lagrange multipliers (for the state-equation) as regular as possible. Nonlinear problems usually involve smooth data. The general duality theory for the mathematical programming in Banach spaces provides Lagrange multipliers in dual spaces. The smoother the spaces for the data, the larger the dual spaces are. This means that, even if we ...

Second order methods for open loop optimal control problems governed by the two-dimensional instationary Navier-Stokes equations are investigated. Optimality systems based on a Lagrangian formulation and adjoint equations are derived. The Newton and quasi-Newton methods as well as various variants of SQP-methods are developed for applications to optimal ow control and their complexity in terms of system solves is discussed. Local convergence and rate of convergence are proved. A numerical example illustrates the feasibility of solving optimal control problems for two-dimensional instationary Navier-Stokes equations by second order numerical methods in a standard workstation environment. Previously such problems were solved by gradient type methods.

Inverse problems of distributed parameter systems with applications to optimal control and identification are considered. Numerical methods and their numerical analysis for solving this kind of inverse problems are presented, main emphasis being on the estimates of the rate of convergence for various schemes. Finally, based on the given error estimates, a two--grid method and related algorithms are introduced, which can be used to solve nonlinear inverse problems effectively. 1 Introduction and preliminaries We consider inverse problems of distributed parameter systems, their numerical approximation with the finite element method, and, especially, introduce estimates of the rate of convergence for inverse identification and optimal control problems. In many applications, inverse problems can be nonlinear and ill-posed which makes them difficult to solve numerically. The parameter-toobservation mapping is often nonlinear and not invertible. Here, University of Jyvaskyla, Department ...

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this paper we discuss the relation between two most commonly used solution methods for output--least--squares formulated nonlinear optimal control problems with bilinear state--constraints. We consider the classical method where one treats the solution of the state equation as dependent variable of the control and the

This paper is concerned with the optimal control of systems governed by a variational inequality coupled with a semilinear partial di#erential equation via the constraint of obstacle. In the stationary case, we consider an elliptic obstacle variational inequality

Formulation In the most general form, we can write an optimization problem in a topological space endowed with some topology and J : R is the objective functional. By extending the objective functional to U via J(u) := we can rewrite this problem as .

Control problems with mixed constraints and application to an optimal investment problem J.F. BONNANS and D. TIBA ∗ Abbreviated title: Control problems with mixed constraints Abstract We discuss two optimal control problems of parabolic equations, with mixed state and control constraints, for which the standard qualification condition does not hold. Our first example is a bottleneck problem, and the second one is an optimal investment problem where a utility type function is to be minimized. By an adapted penalization technique, we derive optimality conditions from which useful information of the solution can be derived. In the case of a control entering linearly in the state equation and cost function, we obtain generalized bang-bang properties.

Distributed optimal control problems for the time-dependent and the stationary Navier-Stokes equations subject to pointwise control constraints are considered. Under a coercivity condition on the Hessian of the Lagrange function, optimal solutions are shown to be directionally differentiable functions of perturbation parameters such as the Reynolds number, the desired trajectory, or the initial conditions. The derivative is characterized as the solution of an auxiliary linear-quadratic optimal control problem. Thus, it can be computed at relatively low cost. Taylor expansions of the minimum value function are provided as well. 1