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PROBLEM FORMULATION FOR MULTIDISCIPLINARY OPTIMIZATION
, 1994
"... This paper is about multidisciplinary (design) optimization, or MDO, the coupling of two or more analysis disciplines with numerical optimization. The paper has three goals. First, it is an expository introduction to MDO aimed at those who do research on optimization algorithms, since the optimizati ..."
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Cited by 112 (8 self)
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This paper is about multidisciplinary (design) optimization, or MDO, the coupling of two or more analysis disciplines with numerical optimization. The paper has three goals. First, it is an expository introduction to MDO aimed at those who do research on optimization algorithms, since the optimization community has much to contribute to this important class of computational engineering problems. Second, this paper presents to the MDO research community a new abstraction for multidisciplinary analysis and design problems as well as new decomposition formulations for these problems. Third, the "individual discipline feasible " (IDF) approaches introduced here make use of existing specialized analysis codes, and they introduce significant opportunities for coarsegrained computational parallelism particularly well suited to heterogeneous computing environments. The key distinguishing characteristic of the three fundamental approaches to MDO formulation discussed here is the kind of disciplinary feasibility that must be maintained at each optimization iteration. Other formulation issues, such as the sensitivities required, are also considered. This discussion highlights the tradeoffs between reuse of existing software, computational requirements, and probability of success.
TrustRegion InteriorPoint Algorithms For Minimization Problems With Simple Bounds
 SIAM J. CONTROL AND OPTIMIZATION
, 1995
"... Two trustregion interiorpoint algorithms for the solution of minimization problems with simple bounds are analyzed and tested. The algorithms scale the local model in a way similar to Coleman and Li [1]. The first algorithm is more usual in that the trust region and the local quadratic model are c ..."
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Cited by 54 (18 self)
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Two trustregion interiorpoint algorithms for the solution of minimization problems with simple bounds are analyzed and tested. The algorithms scale the local model in a way similar to Coleman and Li [1]. The first algorithm is more usual in that the trust region and the local quadratic model are consistently scaled. The second algorithm proposed here uses an unscaled trust region. A global convergence result for these algorithms is given and dogleg and conjugategradient algorithms to compute trial steps are introduced. Some numerical examples that show the advantages of the second algorithm are presented.
TrustRegion InteriorPoint SQP Algorithms For A Class Of Nonlinear Programming Problems
 SIAM J. CONTROL OPTIM
, 1997
"... In this paper a family of trustregion interiorpoint 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 co ..."
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Cited by 43 (9 self)
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In this paper a family of trustregion interiorpoint 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 trustregion techniques for equalityconstrained optimizatio...
Xumerical solution of a flow control problem: vorticity reduction by dynamic boundary action
 Siam Journal in Control and Optimization
, 1998
"... Abstract. In order to laminarize an unsteady, internal flow, the vorticity field is minimized, in a leastsquares sense, using an optimalcontrol approach. The flow model is the Navier–Stokes equation for a viscous incompressible fluid, and the flow is controlled by suction and blowing on a part of ..."
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Cited by 26 (0 self)
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Abstract. In order to laminarize an unsteady, internal flow, the vorticity field is minimized, in a leastsquares sense, using an optimalcontrol approach. The flow model is the Navier–Stokes equation for a viscous incompressible fluid, and the flow is controlled by suction and blowing on a part of the boundary. A quasiNewton method is used for the minimization of a quadratic objective function involving a measure of the vorticity and a regularization term. The Navier–Stokes equations are approximated using a finitedifference scheme in time and finiteelement approximations in space. Accurate expressions for the gradient of the discrete objective function are needed to obtain a satisfactory convergence rate of the minimization algorithm. Therefore, firstorder necessary conditions for a minimizer of the objective function are derived in the fully discrete case. A memorysaving device is discussed without which problems of any realistic size, especially in three space dimensions, would remain computationally intractable. The feasibility of the optimalcontrol approach for flowcontrol problems is demonstrated by numerical experiments for a twodimensional flow in a rectangular cavity at a Reynolds number high enough for nonlinear effects to be important.
Formulation and Analysis of a Sequential Quadratic Programming Method for the Optimal Dirichlet Boundary Control of NavierStokes Flow
, 1997
"... The optimal boundary control of NavierStokes flow is formulated as a constrained optimization problem and a sequential quadratic programming (SQP) approach is studied for its solution. Since SQP methods treat states and controls as independent variables and do not insist on satisfying the constrai ..."
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Cited by 18 (1 self)
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The optimal boundary control of NavierStokes flow is formulated as a constrained optimization problem and a sequential quadratic programming (SQP) approach is studied for its solution. Since SQP methods treat states and controls as independent variables and do not insist on satisfying the constraints during the iterations, care must be taken to avoid a possible incompatibility of Dirichlet boundary conditions and incompressibility constraint. In this paper, compatibility is enforced by choosing appropriate function spaces. The resulting optimization problem is analyzed. Differentiability of the constraints and surjectivity of linearized constraints are verified and adjoints are computed. An SQP method is applied to the optimization problem and compared with other approaches.
A Parallel Reduced Hessian SQP Method for Shape Optimization
"... 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 equation ..."
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Cited by 15 (4 self)
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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 solverboth constructing and solvi...
Optimal Control Of Two And ThreeDimensional Incompressible NavierStokes Flows
, 1997
"... . The focus of this work is on the development of largescale numerical optimization methods for optimal control of steady incompressible NavierStokes 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 w ..."
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Cited by 14 (3 self)
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. The focus of this work is on the development of largescale numerical optimization methods for optimal control of steady incompressible NavierStokes 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 quasiNewton 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 twodimensional flow around a cylinder and threedimensional flow around a sphere. The examples demonstrate at least an orderofmagnitude 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...
A Comparison of Nonlinear Programming Approaches to an Elliptic Inverse Problem and a New Domain Decomposition Approach
, 1994
"... We compare three nonlinear programming approaches to a wellknown elliptic inverse problem in three spatial dimensions. Two of these approaches may be viewed as conventional; the third approach is new and is based on a domain decomposition technique for the solution of the governing elliptic equat ..."
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Cited by 12 (3 self)
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We compare three nonlinear programming approaches to a wellknown elliptic inverse problem in three spatial dimensions. Two of these approaches may be viewed as conventional; the third approach is new and is based on a domain decomposition technique for the solution of the governing elliptic equation. We discuss the benefits that may be obtained from treating the governing differential equation in an inverse problem as equality constraints in the optimization problem. We present numerical results and discuss the relative efficacy of the three approaches.
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...
Formulation and Analysis of a Sequential Quadratic Programming Method for the Optimal Dirichlet Boundary Control of Navier–Stokes Flow
, 1997
"... The optimal boundary control of Navier–Stokes flow is formulated as a constrained optimization problem and a sequential quadratic programming (SQP) approach is studied for its solution. Since SQP methods treat states and controls as independent variables and do not insist on satisfying the constrain ..."
Abstract
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The optimal boundary control of Navier–Stokes flow is formulated as a constrained optimization problem and a sequential quadratic programming (SQP) approach is studied for its solution. Since SQP methods treat states and controls as independent variables and do not insist on satisfying the constraints during the iterations, care must be taken to avoid a possible incompatibility of Dirichlet boundary conditions and incompressibility constraint. In this paper, compatibility is enforced by choosing appropriate function spaces. The resulting optimization problem is analyzed. Differentiability of the constraints and surjectivity of linearized constraints are verified and adjoints are computed. An SQP method is applied to the optimization problem and compared with other approaches.