Journal ArticleDOI
On constrained optimization by adjoint based quasi-Newton methods
Andreas Griewank,Andrea Walther +1 more
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A new approach to constrained optimization that is based on direct and adjoint vector-function evaluations in combination with secant updating is proposed, which avoids the avoidance of constraint Jacobian evaluations and the reduction of the linear algebra cost per iteration in the dense, unstructured case.Abstract:
In this article we propose a new approach to constrained optimization that is based on direct and adjoint vector-function evaluations in combination with secant updating. The main goal is the avoidance of constraint Jacobian evaluations and the reduction of the linear algebra cost per iteration to $ {\cal O}(n + m)^2 $ operations in the dense, unstructured case. A crucial building block is a transformation invariant two-sided-rank-one update (TR1) for approximations to the (active) constraint Jacobian. In this article we elaborate its basic properties and report preliminary numerical results for the new total quasi-Newton approach on some small equality constrained problems. A nullspace implementation under development is briefly described. The tasks of identifying active constraints, safeguarding convergence and many other important issues in constrained optimization are not addressed in detail.read more
Citations
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Journal ArticleDOI
Selected Topics in Column Generation
TL;DR: The growing understanding of the dual point of view is emphasized, which has brought considerable progress to the column generation theory and practice, and is an ever recurring concept in "selected topics."
Book ChapterDOI
Efficient Numerical Methods for Nonlinear MPC and Moving Horizon Estimation
TL;DR: In this article, numerical methods for solving real-time optimization problems in nonlinear model predictive control (NMPC) and moving horizon estimation (MHE) have been reviewed, focusing exclusively on a discrete time setting.
Journal ArticleDOI
Characterization and computation of optimal distributions for channel coding
Jianyi Huang,Sean P. Meyn +1 more
TL;DR: It is shown that the bounds converge to the true channel capacity, and that the distributions converge weakly to a capacity-achieving distribution, along with upper and lower bounds on channel capacity.
Book ChapterDOI
Implementing Mixed Integer Column Generation
TL;DR: This paper summarizes recent work in the field, in particular that of Vanderbeck (2002, 2003), and reviews the main issues that arise when implementing a column generation approach to solve a mixed integer program.
Journal ArticleDOI
Comparison of bundle and classical column generation
Olivier Briant,Claude Lemaréchal,Ph. Meurdesoif,Sophie Michel,Nancy Perrot,François Vanderbeck +5 more
TL;DR: Alternative stabilization techniques used in column generation are reviewed, comparing them from both primal and dual points of view, and a sketchy comparison with the volume algorithm is given.
References
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Book
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
TL;DR: In this paper, Schnabel proposed a modular system of algorithms for unconstrained minimization and nonlinear equations, based on Newton's method for solving one equation in one unknown convergence of sequences of real numbers.
Book
Numerical methods for unconstrained optimization and nonlinear equations
TL;DR: Newton's Method for Nonlinear Equations and Unconstrained Minimization and methods for solving nonlinear least-squares problems with Special Structure.
Book
Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation
Andreas Griewank,Andrea Walther +1 more
TL;DR: This second edition has been updated and expanded to cover recent developments in applications and theory, including an elegant NP completeness argument by Uwe Naumann and a brief introduction to scarcity, a generalization of sparsity.
Journal ArticleDOI
An optimal control approach to a posteriori error estimation in finite element methods
Roland Becker,Rolf Rannacher +1 more
TL;DR: The ‘dual-weighted-residual method’ is introduced initially within an abstract functional analytic setting, and is then developed in detail for several model situations featuring the characteristic properties of elliptic, parabolic and hyperbolic problems.
Book
Linear and Nonlinear Programming
Stephen G. Nash,Ariela Sofer +1 more
TL;DR: This chapter discusses the development of optimization models for constrained computer programming and some of the methods used to achieve this goal were developed in the 1980s and 1990s.