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Title:

Automatic Differentiation: A Structure-Exploiting Forward Mode with Almost Optimal Complexity for Kantorovic Trees

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. A structure-exploiting forward mode is discussed that achieves almost optimal complexity for functions given by Kantorovic trees. It is based on approriate representations of the gradient and the Hessian. After a brief exposition of the forward and reverse mode of automatic differentiation for derivatives up to second order and compact proofs ...

. A structure-exploiting forward mode is discussed that achieves almost optimal complexity for functions given by Kantorovic trees. It is based on approriate representations of the gradient and the Hessian. After a brief exposition of the forward and reverse mode of automatic differentiation for derivatives up to second order and compact proofs of their complexities, the new forward mode is presented and analyzed. It is shown that in the case of functions f : IR n ! IR with a tree as Kantorovic graph the algorithm is only O(ln(n)) times as expensive as the reverse mode. Except for the fact that the new method is a very efficient implementation of the forward mode, it can be used to significantly reduce the length of characterizing sequences before applying the memory expensive reverse mode. For the Hessian all discussed algorithms are shown to be efficiently parallelizable. Some numerical examples confirm the advantages of the new forward mode. Keywords. automatic differentiation, characterizing sequence, code list, forward mode, reverse mode, Kantorovic graph, Kantorovic tree, time complexity, parallelization. 1. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-15

Source:

http://www-m1.mathematik.tu-muenchen.de/m1/personen/mulbrich/papers/TR-IAMS1996-1-TUM.ps.gz

http://www-m1.mathematik.tu-muenchen.de/m1/personen/mulbrich/papers/TR-IAMS1996-1-TUM.ps.gz Minimize

Document Type:

text

Language:

en

Subjects:

characterizing sequence ; code list ; forward mode ; reverse mode ; Kantorovic graph ; Kantorovic tree ; time complexity ; parallelization

characterizing sequence ; code list ; forward mode ; reverse mode ; Kantorovic graph ; Kantorovic tree ; time complexity ; parallelization Minimize

DDC:

511 General principles of mathematics *(computed)*

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Metadata may be used without restrictions as long as the oai identifier remains attached to it.

Metadata may be used without restrictions as long as the oai identifier remains attached to it. Minimize

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Title:

Superlinear Convergence of Affine-Scaling Interior-Point Newton Methods for Infinite-Dimensional Nonlinear Problems with Pointwise Bounds

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We develop and analyze a superlinearly convergent affine-scaling interior-point Newton method for infinite-dimensional 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-dimensional convergence theory by Coleman and Li (SI...

We develop and analyze a superlinearly convergent affine-scaling interior-point Newton method for infinite-dimensional 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-dimensional convergence theory by Coleman and Li (SIAM J. Optim., 6 (1996), pp. 418--445) 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 Newton-like iteration for an affine-scaling formulation of the KKT-condition. 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 . Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-11-21

Source:

http://www-lit.ma.tum.de/veroeff/quel/970.49014.ps.gz

http://www-lit.ma.tum.de/veroeff/quel/970.49014.ps.gz Minimize

Document Type:

text

Language:

en

DDC:

518 Numerical analysis *(computed)*

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Metadata may be used without restrictions as long as the oai identifier remains attached to it.

Metadata may be used without restrictions as long as the oai identifier remains attached to it. Minimize

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Title:

Superlinear Convergence of Affine-Scaling Interior-Point Newton Methods for Infinite-Dimensional Nonlinear Problems with Pointwise Bounds

Description:

We develop and analyze a superlinearly convergent affine-scaling interior-point Newton method for infinite-dimensional 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-dimensional convergence theory by Coleman and Li (SI...

We develop and analyze a superlinearly convergent affine-scaling interior-point Newton method for infinite-dimensional 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-dimensional convergence theory by Coleman and Li (SIAM J. Optim., 6 (1996), pp. 418--445) 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 Newton-like iteration for an affine-scaling formulation of the KKT-condition. 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 . Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2011-09-22

Source:

ftp://softlib.rice.edu/pub/CRPC-TRs/reports/CRPC-TR97697.ps.gz

ftp://softlib.rice.edu/pub/CRPC-TRs/reports/CRPC-TR97697.ps.gz Minimize

Document Type:

text

Language:

en

Subjects:

optimality

optimality Minimize

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Metadata may be used without restrictions as long as the oai identifier remains attached to it.

Metadata may be used without restrictions as long as the oai identifier remains attached to it. Minimize

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Title:

Primal-dual interior-point methods for pde-constrained optimization

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Abstract. This paper provides a detailed analysis of a primal-dual interior-point method for PDE-constrained optimization. Considered are optimal control problems with control constraints in L p. It is shown that the developed primal-dual interior-point method converges globally and locally superlinearly. Not only the easier L ∞-setting is analy...

Abstract. This paper provides a detailed analysis of a primal-dual interior-point method for PDE-constrained optimization. Considered are optimal control problems with control constraints in L p. It is shown that the developed primal-dual interior-point method converges globally and locally superlinearly. Not only the easier L ∞-setting is analyzed, but also a more involved L q-analysis, q < ∞, is presented. In L ∞ , the set of feasible controls contains interior points and the Fréchet differentiability of the perturbed optimality system can be shown. In the L q-setting, which is highly relevant for PDEconstrained optimization, these nice properties are no longer available. Nevertheless, a convergence analysis is developed using refined techniques. In particular, two-norm techniques and a smoothing step are required. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-01-07

Source:

http://www.optimization-online.org/DB_FILE/2006/04/1374.pdf

http://www.optimization-online.org/DB_FILE/2006/04/1374.pdf Minimize

Document Type:

text

Language:

en

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Title:

Superlinear Convergence of Affine-Scaling Interior-Point Newton Methods for Infinite-Dimensional Nonlinear Problems with Pointwise Bounds

Description:

We develop and analyze a superlinearly convergent affine-scaling interior-point Newton method for infinite-dimensional 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-dimensional convergence theory by Coleman and Li (SIAM...

We develop and analyze a superlinearly convergent affine-scaling interior-point Newton method for infinite-dimensional 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-dimensional convergence theory by Coleman and Li (SIAM J. Optim., 6 (1996), pp. 418--445) 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 Newton-like iteration for an affine-scaling formulation of the KKT-condition. 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 . Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2011-05-23

Source:

http://www-m1.ma.tum.de/m1/personen/mulbrich/papers/Local.ps.gz

http://www-m1.ma.tum.de/m1/personen/mulbrich/papers/Local.ps.gz Minimize

Document Type:

text

Language:

en

DDC:

518 Numerical analysis *(computed)*

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Title:

Nonmonotone Trust Region Methods for Nonlinear Equality Constrained Optimization without a Penalty Function

Description:

We propose and analyze a class of penalty-function-free nonmonotone trust-region 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....

We propose and analyze a class of penalty-function-free nonmonotone trust-region 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 Byrd--Omojokun 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 trust-region 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 filter-based 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. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-08-27

Source:

http://www-m1.mathematik.tu-muenchen.de/m1/personen/mulbrich/papers/nmtr.ps.gz

http://www-m1.mathematik.tu-muenchen.de/m1/personen/mulbrich/papers/nmtr.ps.gz Minimize

Document Type:

text

Language:

en

Subjects:

convergence ; equality constraints ; local convergence ; large-scale optimization

convergence ; equality constraints ; local convergence ; large-scale optimization Minimize

DDC:

518 Numerical analysis *(computed)*

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Title:

Superlinear and quadratic convergence of affine-scaling interior-point Newton methods for problems with simple bounds without strict complementarity assumption

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A class of affine-scaling interior-point methods for bound constrained optimization problems is introduced which are locally q--superlinear or q--quadratic convergent. It is assumed that the strong second order sufficient optimality conditions at the solution are satisfied, but strict complementarity is not required. The methods are modification...

A class of affine-scaling interior-point methods for bound constrained optimization problems is introduced which are locally q--superlinear or q--quadratic convergent. It is assumed that the strong second order sufficient optimality conditions at the solution are satisfied, but strict complementarity is not required. The methods are modifications of the affine-scaling interior-point Newton methods introduced by T. F. Coleman and Y. Li (Math. Programming, 67:189--224, 1994). There are two modifications. One is a modification of the scaling matrix, the other one is the use of a projection of the step to maintain strict feasibility rather than a simple scaling of the step. A comprehensive local convergence analysis is given. A simple example is presented to illustrate the pitfalls of the original approach by Coleman and Li in the degenerate case and to demonstrate the performance of the fast converging modifications developed in this paper. Key words: Bound constraints, affine scaling, in. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-13

Source:

http://www.caam.rice.edu/~heinken/papers/deg-local.ps.gz

http://www.caam.rice.edu/~heinken/papers/deg-local.ps.gz Minimize

Document Type:

text

Language:

en

Subjects:

conditions

conditions Minimize

DDC:

510 Mathematics *(computed)*

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Title:

Nonmonotone trust region methods for nonlinear equality constrained optimization without a penalty function

Author:

Description:

Abstract. We propose and analyze a class of penalty-function-free nonmonotone trust-region 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...

Abstract. We propose and analyze a class of penalty-function-free nonmonotone trust-region 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 Byrd–Omojokun 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 trust-region 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 filter-based 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. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-07-17

Source:

http://www-lit.ma.tum.de/veroeff/quel/009.65011.pdf

http://www-lit.ma.tum.de/veroeff/quel/009.65011.pdf Minimize

Document Type:

text

Language:

en

Subjects:

Key words

Key words Minimize

DDC:

518 Numerical analysis *(computed)*

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Title:

A globally convergent primal-dual interior-point filter method for nonlinear programming

Author:

Description:

A globally convergent primal-dual interior-point filter method for nonlinear programming

A globally convergent primal-dual interior-point filter method for nonlinear programming Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-07-17

Source:

http://www.opt.tu-darmstadt.de/forschung/nichtlin/Team/ulbrich/papers/ipfilter.pdf

http://www.opt.tu-darmstadt.de/forschung/nichtlin/Team/ulbrich/papers/ipfilter.pdf Minimize

Document Type:

text

Language:

en

Subjects:

Key words. interior-point methods ; primal-dual ; filter

Key words. interior-point methods ; primal-dual ; filter Minimize

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Title:

Semismooth Newton methods for operator equations in function spaces

Description:

We develop a semismoothness concept for nonsmooth superposition operators in function spaces. The considered class of operators includes NCP-function-based reformulations of infinite-dimensional nonlinear complementarity problems, and thus covers a very comprehensive class of applications. Our results generalize semismoothness and ff-order semis...

We develop a semismoothness concept for nonsmooth superposition operators in function spaces. The considered class of operators includes NCP-function-based reformulations of infinite-dimensional nonlinear complementarity problems, and thus covers a very comprehensive class of applications. Our results generalize semismoothness and ff-order semismoothness from finite-dimensional spaces to a Banach space setting. Hereby, a new generalized differential is used that can be seen as an extension of Qi's finite-dimensional C-subdifferential to our infinite-dimensional framework. We apply these semismoothness results to develop a Newton-like method for nonsmooth operator equations and prove its local q-superlinear convergence to regular solutions. If the underlying operator is ff-order semismoothness, convergence of q-order 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. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-10-05

Source:

http://www.caam.rice.edu/caam/trs/2000/TR00-11.ps

http://www.caam.rice.edu/caam/trs/2000/TR00-11.ps Minimize

Document Type:

text

Language:

en

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