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15
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 35 (8 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...
Analysis of Inexact TrustRegion SQP Algorithms
 RICE UNIVERSITY, DEPARTMENT OF
, 2000
"... In this paper we extend the design of a class of compositestep trustregion SQP methods and their global convergence analysis to allow inexact problem information. The inexact problem information can result from iterative linear systems solves within the trustregion SQP method or from approximatio ..."
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Cited by 17 (2 self)
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In this paper we extend the design of a class of compositestep trustregion SQP methods and their global convergence analysis to allow inexact problem information. The inexact problem information can result from iterative linear systems solves within the trustregion SQP method or from approximations of firstorder derivatives. Accuracy requirements in our trustregion SQP methods are adjusted based on feasibility and optimality of the iterates. Our accuracy requirements are stated in general terms, but we show how they can be enforced using information that is already available in matrixfree implementations of SQP methods. In the absence of inexactness our global convergence theory is equal to that of Dennis, ElAlem, Maciel (SIAM J. Optim., 7 (1997), pp. 177207). If all iterates are feasible, i.e., if all iterates satisfy the equality constraints, then our results are related to the known convergence analyses for trustregion methods with inexact gradient information fo...
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 15 (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.
An Optimal Control Problem for Flows with Discontinuities
, 1996
"... In this paper we study a design problem for a duct flow with a shock. The presence of the shock causes numerical difficulties. Good shockcapturing schemes with low continuity properties often cannot be combined successfully with efficient optimization methods requiring smooth functions. A remedy st ..."
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Cited by 13 (5 self)
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In this paper we study a design problem for a duct flow with a shock. The presence of the shock causes numerical difficulties. Good shockcapturing schemes with low continuity properties often cannot be combined successfully with efficient optimization methods requiring smooth functions. A remedy studied in this paper is to introduce the shock location as an explicit variable. This allows one to fit the shock and yields a problem with sufficiently smooth functions. We prove the existence of optimal solutions, Frechet differentiability, and the existence of Lagrange multipliers. In the second part we introduce and investigate the discrete problem and study the relations between the optimality conditions for the infinite dimensional problem and the discretized one. This reveals information important for the numerical solution of the problem. Numerical examples are given to demonstrate the theoretical findings.
An Interface Between Optimization and Application for the Numerical Solution of Optimal Control Problems
 ACM Transactions on Mathematical Software
, 1998
"... This paper is concerned with the implementation of optimization algorithms for the solution of smooth discretized optimal control problems. The problems under consideration can be written as min f(y; u) ..."
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Cited by 13 (7 self)
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This paper is concerned with the implementation of optimization algorithms for the solution of smooth discretized optimal control problems. The problems under consideration can be written as min f(y; u)
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...
Airfoil Design by an AllAtOnce Method
, 1997
"... The allatonce approach is implemented to solve an optimum airfoil design problem. The airfoil design problem is formulated as a constrained optimization problem in which flow variables and design variables are viewed as independent and the coupling steady state Euler equation is included as a cons ..."
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Cited by 9 (0 self)
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The allatonce approach is implemented to solve an optimum airfoil design problem. The airfoil design problem is formulated as a constrained optimization problem in which flow variables and design variables are viewed as independent and the coupling steady state Euler equation is included as a constraint, along with geometry and other constraints. In this formulation, the optimizer computes a sequence of points which tend toward feasiblility and optimality at the same time (allatonce). This decoupling of variables typically makes the problem less nonlinear and can lead to more efficient solutions. In this paper an existing optimization algorithm is combined with an existing flow code. The problem formulation, its discretization, and the underlying solvers are described. Implementation issues are presented and numerical results are given which indicate that the cost of solving the design problem is approximately six times the cost of solving a single analysis problem.
On the Convergence Theory of TrustRegionBased Algorithms for EqualityConstrained Optimization
, 1995
"... In this paper we analyze incxact trust region interior point (TRIP) sequential quadr tic programming (SOP) 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 8 (0 self)
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In this paper we analyze incxact trust region interior point (TRIP) sequential quadr tic programming (SOP) 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 nonhnear constraints often come from the discretization of partial differential equations. In such cases the calculation of derivative information and the solution of hncarizcd equations is expensive. Often, the solution of hncar systems and derivatives arc computed incxactly yielding nonzero residuals. This paper
A Nonsmooth Equation Based BFGS Method for Solving KKT Systems in Mathematical Programming
 Journal of Optimization Theory and Applications
, 1998
"... In this paper, we present a BFGS method for solving a KKT system in mathematical programming, based on a nonsmooth equation reformulation of the KKT system. We successively split the nonsmooth equation into equivalent equations with particular structure. Based on the splitting, we develop a BFGS met ..."
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Cited by 2 (1 self)
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In this paper, we present a BFGS method for solving a KKT system in mathematical programming, based on a nonsmooth equation reformulation of the KKT system. We successively split the nonsmooth equation into equivalent equations with particular structure. Based on the splitting, we develop a BFGS method in which subproblems are systems of linear equations with symmetric and positive definite coefficient matrices. A suitable line search is introduced under which the generated iterates exhibit an approximately norm decent property. The method is well defined and, under suitable conditions, converges to a KKT point globally and superlinearly without convexity assumption on the problem.
Multilevel algorithms for largescale interior point methods in bound constrained optimization, tech
, 2006
"... Abstract. We develop and compare multilevel algorithms for solving bound constrained nonlinear variational problems via interior point methods. Several equivalent formulations of the linear systems arising at each iteration of the interior point method are compared from the point of view of conditio ..."
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Cited by 2 (0 self)
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Abstract. We develop and compare multilevel algorithms for solving bound constrained nonlinear variational problems via interior point methods. Several equivalent formulations of the linear systems arising at each iteration of the interior point method are compared from the point of view of conditioning and iterative solution. Furthermore, we show how a multilevel continuation strategy can be used to obtain good initial guesses (“hot starts”) for each nonlinear iteration. A minimal surface problem is used to illustrate the various approaches. 1. Introduction. In