Results 1  10
of
50
A semismooth Newton method for Tikhonov functionals with sparsity constraints
, 2007
"... ..."
(Show Context)
Semismooth Newton methods for variational inequalities of the first kind
, 2003
"... Semi–smooth Newton methods are analyzed for a class of variational inequalities in infinite dimensions. It is shown that they are equivalent to certain active set strategies. Global and local superlinear convergence are proved. To overcome the phenomenon of finite speed of propagation of discreti ..."
Abstract

Cited by 27 (9 self)
 Add to MetaCart
(Show Context)
Semi–smooth Newton methods are analyzed for a class of variational inequalities in infinite dimensions. It is shown that they are equivalent to certain active set strategies. Global and local superlinear convergence are proved. To overcome the phenomenon of finite speed of propagation of discretized problems a penalty version is used as the basis for a continuation procedure to speed up convergence. The choice of the penalty parameter can be made on the basis of an L ∞ estimate for the penalized solutions. Unilateral as well as bilateral problems are considered.
A dualitybased approach to elliptic control problems in nonreflexive Banach spaces
, 2009
"... ..."
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 ..."
Abstract

Cited by 21 (3 self)
 Add to MetaCart
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
PathFollowing Methods for a Class of Constrained Minimization Problems in Function Space
, 2004
"... Pathfollowing methods for primaldual active set strategies requiring a regularization parameter are introduced. Existence of a path and its differentiability properties are analyzed. Monotonicity and convexity of the primaldual path value function are investigated. Both feasible and infeasible ap ..."
Abstract

Cited by 21 (5 self)
 Add to MetaCart
(Show Context)
Pathfollowing methods for primaldual active set strategies requiring a regularization parameter are introduced. Existence of a path and its differentiability properties are analyzed. Monotonicity and convexity of the primaldual path value function are investigated. Both feasible and infeasible approximations are considered. Numerical path following strategies are developed and their efficiency is demonstrated by means of examples.
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 ..."
Abstract

Cited by 13 (4 self)
 Add to MetaCart
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.
Nonsmooth Newtonlike Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces
, 2002
"... ..."
ElasticNet Regularization: Error estimates and Active Set Methods
, 905
"... This paper investigates theoretical properties and efficient numerical algorithms for the socalled elasticnet regularization originating from statistics, which enforces simultaneously ℓ 1 and ℓ 2 regularization. The stability of the minimizer and its consistency are studied, and convergence rates ..."
Abstract

Cited by 12 (6 self)
 Add to MetaCart
(Show Context)
This paper investigates theoretical properties and efficient numerical algorithms for the socalled elasticnet regularization originating from statistics, which enforces simultaneously ℓ 1 and ℓ 2 regularization. The stability of the minimizer and its consistency are studied, and convergence rates for both a priori and a posteriori parameter choice rules are established. Two iterative numerical algorithms of active set type are proposed, and their convergence properties are discussed. Numerical results are presented to illustrate the features of the functional and algorithms. 1
Lagrange multiplier approach with optimized finite difference stencils for pricing American options under stochastic volatility
 SIAM J. Sci. Comput
"... Abstract. The deterministic numerical valuation of American options under Heston’s stochastic volatility model is considered. The prices are given by a linear complementarity problem with a twodimensional parabolic partial differential operator. A new truncation of the domain is described for small ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
(Show Context)
Abstract. The deterministic numerical valuation of American options under Heston’s stochastic volatility model is considered. The prices are given by a linear complementarity problem with a twodimensional parabolic partial differential operator. A new truncation of the domain is described for small asset values while for large asset values and variance a standard truncation is used. The finite difference discretization is constructed by numerically solving quadratic optimization problem aiming to minimize the truncation error at each grid point. A Lagrange approach is used to treat the linear complementarity problems. Numerical examples demonstrate the accuracy and effectiveness of the proposed approach. Key words. American option pricing, stochastic volatility model, linear complementarity problem, finite difference method, quadratic programming, multigrid method, Lagrange method, penalty method
Semismooth Newton and augmented Lagrangian methods for a simplified friction problem
 SIAM J. Optim
"... Abstract. In this paper a simplified friction problem and iterative secondorder algorithms for its solution are analyzed in infinite dimensional function spaces. Motivated from the dual formulation, a primaldual active set strategy and a semismooth Newton method for a regularized problem as well a ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper a simplified friction problem and iterative secondorder algorithms for its solution are analyzed in infinite dimensional function spaces. Motivated from the dual formulation, a primaldual active set strategy and a semismooth Newton method for a regularized problem as well as an augmented Lagrangian method for the original problem are presented and their close relation is analyzed. Local as well as global convergence results are given. By means of numerical tests, we discuss among others convergence properties, the dependence on the mesh, and the role of the regularization and illustrate the efficiency of the proposed methodologies.