Results 1  10
of
21
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 50 (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...
Recursive trustregion methods for multiscale nonlinear optimization
 SIAM J. OPTIM
, 2006
"... A class of trustregion methods is presented for solving unconstrained nonlinear and possibly nonconvex discretized optimization problems, like those arising in systems governed by partial differential equations. The algorithms in this class make use of the discretization level as a mean of speedi ..."
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Cited by 18 (3 self)
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A class of trustregion methods is presented for solving unconstrained nonlinear and possibly nonconvex discretized optimization problems, like those arising in systems governed by partial differential equations. The algorithms in this class make use of the discretization level as a mean of speeding up the computation of the step. This use is recursive, leading to true multilevel/multiscale optimization methods reminiscent of multigrid methods in linear algebra and the solution of partialdifferential equations. A simple algorithm of the class is then described and its numerical performance is shown to be numerically promising. This observation then motivates a proof of global convergence to firstorder stationary points on the fine grid that is valid for all algorithms in the class.
Nonmonotone Trust Region Methods for Nonlinear Equality Constrained Optimization without a Penalty Function
 MATH. PROGRAM., SER. B
, 2000
"... We propose and analyze a class of penaltyfunctionfree nonmonotone trustregion methods for nonlinear equality constrained optimization problems. The algorithmic framework yields global convergence without using a merit function and allows nonmonotonicity independently for both, the constraint viol ..."
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Cited by 15 (6 self)
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We propose and analyze a class of penaltyfunctionfree nonmonotone trustregion methods for nonlinear equality constrained optimization problems. The algorithmic framework yields global convergence without using a merit function and allows nonmonotonicity independently for both, the constraint violation and the value of the Lagrangian function. Similar to the ByrdOmojokun class of algorithms, each step is composed of a quasinormal and a tangential step. Both steps are required to satisfy a decrease condition for their respective trustregion subproblems. The proposed mechanism for accepting steps combines nonmonotone decrease conditions on the constraint violation and/or the Lagrangian function, which leads to a flexibility and acceptance behavior comparable to filterbased methods. We establish the global convergence of the method. Furthermore, transition to quadratic local convergence is proved. Numerical tests are presented that confirm the robustness and efficiency of the approach.
Strictly Feasible EquationBased Methods For Mixed Complementarity Problems
, 1999
"... We introduce a new algorithm for the solution of the mixed complementarity problem (MCP) which has stronger properties than most existing methods. In fact, typical solution methods for the MCP either generate feasible iterates but have to solve relatively complicated subproblems (like quadratic pro ..."
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Cited by 15 (6 self)
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We introduce a new algorithm for the solution of the mixed complementarity problem (MCP) which has stronger properties than most existing methods. In fact, typical solution methods for the MCP either generate feasible iterates but have to solve relatively complicated subproblems (like quadratic programs or linear complementarity problems), or they have relatively simple subproblems (like linear systems of equations) but generate not necessarily feasible iterates. The method to be presented here combines the nice features of these two classes of methods: It has to solve only one linear system of equations (of reduced dimension) at each iteration, and it generates feasible (more precisely: strictly feasible) iterates. The new method has some nice global and local convergence properties. Some preliminary numerical results will also be given.
Constrained Optimal Control of NavierStokes Flow by Semismooth Newton Methods
 SYSTEMS & CONTROL LETTERS
, 2002
"... We propose and analyze a semismooth Newtontype method for the solution of a pointwise constrained optimal control problem governed by the timedependent incompressible NavierStokes equations. The method is based on a reformulation of the optimality system as an equivalent nonsmooth operator equati ..."
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Cited by 9 (2 self)
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We propose and analyze a semismooth Newtontype method for the solution of a pointwise constrained optimal control problem governed by the timedependent incompressible NavierStokes equations. The method is based on a reformulation of the optimality system as an equivalent nonsmooth operator equation. We analyze the flow control problem and establish qsuperlinear convergence of the method. In the numerical implementation, adjoint techniques are combined with a truncated conjugate gradient method. Numerical results are presented that support our theoretical results and confirm the viability of the approach.
An Active SetType Newton Method For Constrained Nonlinear Systems
 In Complementarity: Applications, Algorithms and Extensions (2001), Ferris M., Mangasarian O., Pang J.S., (Eds
, 1999
"... . We consider the problem of finding a solution of a nonlinear system of equations subject to some box constraints. To this end, we introduce a new active settype Newton method with global and local fast convergence properties. The method generates feasible iterates and has to solve only one linear ..."
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Cited by 6 (2 self)
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. We consider the problem of finding a solution of a nonlinear system of equations subject to some box constraints. To this end, we introduce a new active settype Newton method with global and local fast convergence properties. The method generates feasible iterates and has to solve only one linear system of equations at each iteration. Due to our active set strategy, this linear system is of reduced dimension. Key Words. Nonlinear equations, box constraints, Newton's method, active set strategy, projected gradient, global convergence, quadratic convergence. 1 The research of this author was partially supported by the DFG (Deutsche Forschungsgemeinschaft). 1 Introduction The problem we address in this paper is to find a solution of the constrained nonlinear system F (x) = 0; x 2 [l; u]; (1) where F : [l; u] ! IR n is a given function which is assumed to be continuously differentiable in an open set containing the box [l; u], and where l = (l 1 ; : : : ; l n ) T ; u = (u 1 ; ...
2004): On a semismooth least squares formulation of complementarity problems with gap reduction
 Optim. Meth. Soft
"... Abstract: We present a nonsmooth least squares reformulation of the complementarity problem and investigate its convergence properties. The global and local fast convergence results (under mild assumptions) are similar to some existing equationbased methods. In fact, our least squares formulation i ..."
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Cited by 5 (1 self)
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Abstract: We present a nonsmooth least squares reformulation of the complementarity problem and investigate its convergence properties. The global and local fast convergence results (under mild assumptions) are similar to some existing equationbased methods. In fact, our least squares formulation is obtained by modifying one of these equationbased methods (using the FischerBurmeister function) in such a way that we overcome a major drawback of this equationbased method. The resulting nonsmooth LevenbergMarquardttype method turns out to be significantly more robust than the corresponding equationbased method. This is illustrated by our numerical results using the MCPLIB test problem collection.
Projected filter trust region methods for a semismooth leastsquares formulation of mixed complementarity problems
 Optim. Method Soft
"... Abstract: A reformulation of the mixed complementarity problem as a box constrained overdetermined system of semismooth equations or, equivalently, a box constrained nonlinear least squares problem with zero residual is presented. Based on this reformulation, a trust region method for the solution ..."
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Cited by 5 (0 self)
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Abstract: A reformulation of the mixed complementarity problem as a box constrained overdetermined system of semismooth equations or, equivalently, a box constrained nonlinear least squares problem with zero residual is presented. Based on this reformulation, a trust region method for the solution of mixed complementarity problems is considered. This trust region method contains elements from different areas: A projected LevenbergMarquardt step in order to guarantee local fast convergence under suitable assumptions, affine scaling matrices which are used to improve the global convergence properties, and a multidimensional filter technique to accept a full step more frequently. Global convergence results as well as local superlinear/quadratic convergence is shown under appropriate assumptions. Moreover, numerical results for the MCPLIB indicate that the overall method is quite robust.
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
Solving KKT Systems via the Trust Region and the Conjugate Gradient Methods
 Applied Mathematics Research Report AMR99/19, School of Mathematics, University of New South
, 2000
"... : A popular approach to solve the KarushKuhnTucker (KKT) system, mainly arising from the variational inequality problem and the constrained optimization problem, is to reformulate it as a constrained minimization problem with simple bounds. In this paper, we propose a trust region method for solvi ..."
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Cited by 2 (1 self)
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: A popular approach to solve the KarushKuhnTucker (KKT) system, mainly arising from the variational inequality problem and the constrained optimization problem, is to reformulate it as a constrained minimization problem with simple bounds. In this paper, we propose a trust region method for solving the reformulation problem with the trust region subproblem being solved by the truncated conjugate gradient (CG) method, which is cost eective. Other advantages of the proposed method over the existing ones include that a good approximation solution to the trust region subproblem can be found by the truncated CG method and is judged in a simple way; and that the working matrix in each iteration is H, instead of the condensed H T H, where H is a matrix element of the generalized Jacobian of the function used in the reformulation. As a matter of fact, the matrix used is of reduced dimension. Extra attention is taken to ensure the success of the truncated CG method as well as the feasibil...