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55
Newton's Method For Large BoundConstrained Optimization Problems
 SIAM JOURNAL ON OPTIMIZATION
, 1998
"... We analyze a trust region version of Newton's method for boundconstrained problems. Our approach relies on the geometry of the feasible set, not on the particular representation in terms of constraints. The convergence theory holds for linearlyconstrained problems, and yields global and super ..."
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Cited by 107 (5 self)
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We analyze a trust region version of Newton's method for boundconstrained problems. Our approach relies on the geometry of the feasible set, not on the particular representation in terms of constraints. The convergence theory holds for linearlyconstrained problems, and yields global and superlinear convergence without assuming neither strict complementarity nor linear independence of the active constraints. We also show that the convergence theory leads to an efficient implementation for large boundconstrained problems.
An interior point algorithm for large scale nonlinear programming
, 1997
"... The design and implementation of a new algorithm for solving large nonlinear programming problems is described. It follows a barrier approach that employs sequential quadratic programming and trust regions to solve the subproblems occurring in the iteration. Both primal and primaldual versions of t ..."
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Cited by 88 (18 self)
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The design and implementation of a new algorithm for solving large nonlinear programming problems is described. It follows a barrier approach that employs sequential quadratic programming and trust regions to solve the subproblems occurring in the iteration. Both primal and primaldual versions of the algorithm are developed, and their performance is illustrated in a set of numerical tests.
On the solution of equality constrained quadratic programming problems arising . . .
, 1998
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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 45 (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...
A new active set algorithm for box constrained optimization,SIAM
 Journal on Optimization
, 2006
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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 26 (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...
Superlinear and Quadratic Convergence of AffineScaling InteriorPoint Newton Methods for Problems with Simple Bounds without Strict Complementarity Assumption
, 1998
"... A class of affinescaling interiorpoint methods for bound constrained optimization problems is introduced which are locally qsuperlinear or qquadratic convergent. It is assumed that the strong... ..."
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Cited by 21 (3 self)
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A class of affinescaling interiorpoint methods for bound constrained optimization problems is introduced which are locally qsuperlinear or qquadratic convergent. It is assumed that the strong...
An Adaptive Algorithm for Bound Constrained Quadratic Minimization
, 1997
"... A general algorithm for minimizing a quadratic function with bounds on the variables is presented. The new algorithm can use different unconstrained minimization techniques on different faces. At every face, the minimization technique can be chosen according to he structure of the Hessian and the di ..."
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Cited by 18 (9 self)
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A general algorithm for minimizing a quadratic function with bounds on the variables is presented. The new algorithm can use different unconstrained minimization techniques on different faces. At every face, the minimization technique can be chosen according to he structure of the Hessian and the dimension of the face. The strategy for leaving the face is based on a simple scheme that exploits the properties of the "chopped gradient" introduced by Friedlander and Mart'inez in 1989. This strategy guarantees global convergence even in the presence of dual degeneracy, and finite identification in the nondegenerate case. A slight modification of the algorithm satisfies, in addition, an identification property in the case of dual degeneracy. Numerical experiments combining this new strategy with conjugate gradients, gradient with retards and direct solvers are presented. Key words. Quadratic programming, conjugate gradients, gradient with retards, active set methods, sparse Cholesky factor...
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 17 (0 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.
Superlinear Convergence of AffineScaling InteriorPoint Newton Methods for InfiniteDimensional Nonlinear Problems with Pointwise Bounds
, 1999
"... We develop and analyze a superlinearly convergent affinescaling interiorpoint Newton method for infinitedimensional problems with pointwise bounds in L p space. The problem formulation is motivated by optimal control problems with L p controls and pointwise control constraints. The finite ..."
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Cited by 17 (6 self)
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We develop and analyze a superlinearly convergent affinescaling interiorpoint Newton method for infinitedimensional problems with pointwise bounds in L p space. The problem formulation is motivated by optimal control problems with L p controls and pointwise control constraints. The finitedimensional convergence theory by Coleman and Li (SIAM J. Optim., 6 (1996), pp. 418445) makes essential use of the equivalence of norms and the exact identifiability of the active constraints close to an optimizer with strict complementarity. Since these features are not available in our infinitedimensional framework, algorithmic changes are necessary to ensure fast local convergence. The main building block is a Newtonlike iteration for an affinescaling formulation of the KKTcondition. We demonstrate in an example that a stepsize rule to obtain an interior iterate may require very small stepsizes even arbitrarily close to a nondegenerate solution. Using a pointwise projection instead ...