Results 1  10
of
17
Semismooth Newton methods for operator equations in function spaces
, 2000
"... We develop a semismoothness concept for nonsmooth superposition operators in function spaces. The considered class of operators includes NCPfunctionbased reformulations of infinitedimensional nonlinear complementarity problems, and thus covers a very comprehensive class of applications. Our resul ..."
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Cited by 48 (3 self)
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We develop a semismoothness concept for nonsmooth superposition operators in function spaces. The considered class of operators includes NCPfunctionbased reformulations of infinitedimensional nonlinear complementarity problems, and thus covers a very comprehensive class of applications. Our results generalize semismoothness and fforder semismoothness from finitedimensional spaces to a Banach space setting. Hereby, a new generalized differential is used that can be seen as an extension of Qi's finitedimensional Csubdifferential to our infinitedimensional framework. We apply these semismoothness results to develop a Newtonlike method for nonsmooth operator equations and prove its local qsuperlinear convergence to regular solutions. If the underlying operator is fforder semismoothness, convergence of qorder 1 + ff is proved. We also establish the semismoothness of composite operators and develop corresponding chain rules. The developed theory is accompanied by illustrating e...
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...
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...
Interior point methods in function space
 SIAM J. Control Optim
"... Abstract. A primaldual interior point method for optimal control problems is considered. The algorithm is directly applied to the infinitedimensional problem. Existence and convergence of the central path are analyzed, and linear convergence of a shortstep pathfollowing method is established. Ke ..."
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Cited by 21 (3 self)
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Abstract. A primaldual interior point method for optimal control problems is considered. The algorithm is directly applied to the infinitedimensional problem. Existence and convergence of the central path are analyzed, and linear convergence of a shortstep pathfollowing method is established. Key words. interior point methods in function space, optimal control, complementarity functions
NonMonotone TrustRegion Methods for BoundConstrained Semismooth Equations with Applications to Nonlinear Mixed Complementarity Problems
, 1999
"... We develop and analyze a class of trustregion methods for boundconstrained semismooth systems of equations. The algorithm is based on a simply constrained differentiable minimization reformulation. Our global convergence results are developed in a very general setting that allows for nonmonotoni ..."
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Cited by 21 (4 self)
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We develop and analyze a class of trustregion methods for boundconstrained semismooth systems of equations. The algorithm is based on a simply constrained differentiable minimization reformulation. Our global convergence results are developed in a very general setting that allows for nonmonotonicity of the function values at subsequent iterates. We propose a way of computing trial steps by a semismooth Newtonlike method that is augmented by a projection onto the feasible set. Under a DennisMoretype condition we prove that close to a BDregular solution the trustregion algorithm turns into this projected Newton method, which is shown to converge locally qsuperlinearly or quadratically, respectively, depending on the quality of the approximate BDsubdifferentials used. As an important application we discuss in detail how the developed algorithm can be used to solve nonlinear mixed complementarity problems (MCPs). Hereby, the MCP is converted into a boundconstrained semismooth...
Superlinear convergence of the control reduced interior point method for PDE constrained optimization
 COMP. OPT. AND APPL
, 2005
"... A thorough convergence analysis of the Control Reduced Interior Point Method in function space is performed. This recently proposed method is a primal interior point pathfollowing scheme with the special feature, that the control variable is eliminated from the optimality system. Apart from global l ..."
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Cited by 13 (4 self)
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A thorough convergence analysis of the Control Reduced Interior Point Method in function space is performed. This recently proposed method is a primal interior point pathfollowing scheme with the special feature, that the control variable is eliminated from the optimality system. Apart from global linear convergence we show, that this method converges locally almost quadratically, if the optimal solution satises a certain nondegeneracy condition. In numerical experiments we observe, that a prototype implementation of our method behaves as predicted by our theoretical results.
On affinescaling interiorpoint Newton methods for nonlinear minimization with bound constraints
 Computational Optimization and Applications
"... Abstract. A class of new affinescaling interiorpoint Newtontype methods are considered for the solution of optimization problems with bound constraints. The methods are shown to be locally quadratically convergent under the strong second order sufficiency condition without assuming strict compl ..."
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Cited by 9 (2 self)
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Abstract. A class of new affinescaling interiorpoint Newtontype methods are considered for the solution of optimization problems with bound constraints. The methods are shown to be locally quadratically convergent under the strong second order sufficiency condition without assuming strict complementarity of the solution. The new methods differ from previous ones by Coleman and Li [Mathematical Programming, 67 (1994), pp. 189– 224] and Heinkenschloss, Ulbrich, and Ulbrich [Mathematical Programming, 86 (1999), pp. 615–635] mainly in the choice of the scaling matrix. The scaling matrices used here have stronger smoothness properties and allow the application of standard results from nonsmooth analysis in order to obtain a relatively short and elegant local convergence result. An important tool for the definition of the new scaling matrices is the correct identification of the degenerate indices. Some illustrative numerical results with a comparison of the different scaling techniques are also included. Key Words. Newton’s method, affine scaling, interiorpoint method, quadratic convergence, identification of active constraints. 1
A meshindependence for semismooth NEWTON METHODS
, 2003
"... For a class of semismooth operator equations a mesh independence result for generalized Newton methods is established. The main result states that the continuous and the discrete Newton process, when initialized properly, converge qlinearly with the same rate. The problem class considered in the ..."
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Cited by 7 (1 self)
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For a class of semismooth operator equations a mesh independence result for generalized Newton methods is established. The main result states that the continuous and the discrete Newton process, when initialized properly, converge qlinearly with the same rate. The problem class considered in the paper includes MCPfunction based reformulations of first order conditions of a class of control constrained optimal control problems for partial differential equations for which a numerical validation of the theoretical results is given.
An interiorpoint affinescaling trustregion method for semismooth equations with box constraints
 Comput. Optim. Appl
"... Abstract. An algorithm for the solution of a semismooth system of equations with box constraints is described. The method is an affinescaling trustregion method. All iterates generated by this method are strictly feasible. In this way, possible domain violations outside or on the boundary of the b ..."
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Cited by 3 (0 self)
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Abstract. An algorithm for the solution of a semismooth system of equations with box constraints is described. The method is an affinescaling trustregion method. All iterates generated by this method are strictly feasible. In this way, possible domain violations outside or on the boundary of the box are avoided. The method is shown to have strong global and local convergence properties under suitable assumptions, in particular, when the method is used with a special scaling matrix. Numerical results are presented for a number of problems arising from different areas. Key Words. Affine scaling, trustregion method, nonlinear equations, box constraints, semismooth functions, Newton’s method. 1