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Analysis of Inexact TrustRegion InteriorPoint SQP Algorithms
, 1995
"... In this paper we analyze inexact trustregion interiorpoint (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 applicati ..."
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Cited by 11 (7 self)
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In this paper we analyze inexact trustregion interiorpoint (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 trustregion radius, t...
Second Order Methods For Optimal Control Of TimeDependent Fluid Flow
, 1999
"... Second order methods for open loop optimal control problems governed by the twodimensional instationary NavierStokes equations are investigated. Optimality systems based on a Lagrangian formulation and adjoint equations are derived. The Newton and quasiNewton methods as well as various variants o ..."
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Cited by 8 (3 self)
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Second order methods for open loop optimal control problems governed by the twodimensional instationary NavierStokes equations are investigated. Optimality systems based on a Lagrangian formulation and adjoint equations are derived. The Newton and quasiNewton methods as well as various variants of SQPmethods 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 twodimensional instationary NavierStokes equations by second order numerical methods in a standard workstation environment. Previously such problems were solved by gradient type methods.
Optimality Conditions And Generalized BangBang Principle For A StateConstrained Semilinear Parabolic Problem
 Numerical Functional Analysis and Optimization
, 1996
"... 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 twonorm technique. The theory i ..."
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Cited by 5 (4 self)
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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 twonorm technique. The theory is applied to derive a generalized bangbang principle. 1 This work was supported by EEC, HCM Contract CHRXCT940471 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 stateequation) 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 ...
Numerical Methods for Nonlinear Inverse Problems
, 1996
"... 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 variou ..."
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Cited by 1 (1 self)
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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 twogrid 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 illposed which makes them difficult to solve numerically. The parametertoobservation mapping is often nonlinear and not invertible. Here, University of Jyvaskyla, Department ...
Modeling And Minimization Of Extinction In VolterraLotka Type Equations With Free Boundaries
, 1996
"... An equation of the distributed VolterraLotka type, with free boundary of the obstacle type, with possible applications in ecology, when extinction of the biological species is of particular concern, is introduced and solved. Optimal control problem for such an equation, and in particular the proble ..."
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Cited by 1 (0 self)
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An equation of the distributed VolterraLotka type, with free boundary of the obstacle type, with possible applications in ecology, when extinction of the biological species is of particular concern, is introduced and solved. Optimal control problem for such an equation, and in particular the problem of minimization of the area of extinction of the species, is introduced and to some extent solved. 1 Introduction Modeling of distributed population dynamics was studied in the literature (see [14] and the references given there). The optimal control of distributed population dynamics, by the method of monotone iterations 1 , was introduced by the author and others in [17, 8, 9] 2 . The crucial assumption in those studies is that the harvesting rate of the species is linear with respect to the size of the population. The consequence of such an assumption in a model is that the species is either totally extinct, or its density is strictly positive in all of the considered region. The p...
Some Relations between Two Strategies for Solving Optimal Control Problems with Bilinear Constraints
"... this paper we discuss the relation between two most commonly used solution methods for outputleastsquares formulated nonlinear optimal control problems with bilinear stateconstraints. We consider the classical method where one treats the solution of the state equation as dependent variable of ..."
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this paper we discuss the relation between two most commonly used solution methods for outputleastsquares formulated nonlinear optimal control problems with bilinear stateconstraints. We consider the classical method where one treats the solution of the state equation as dependent variable of the control and the
Indirect Obstacle Control Problem for Variational Inequalities
"... 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 ..."
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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
Infinitedimensional Optimization and Optimal Design
, 2003
"... 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 . ..."
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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 .
Dedicated to Dr. Constantin Vârsan on the occasion of his 70th Birthday
"... 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 ..."
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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 bangbang properties.