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Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods [article]

Michael B. Cohen, Aleksander Madry, Dimitris Tsipras, Adrian Vladu
2017 arXiv   pre-print
We complement our results by providing a separate algorithm that uses an interior-point method and runs in time O(m^3/2 (1/ϵ)).  ...  We then show that in the context of the specific functions capturing matrix scaling and balancing, we can leverage and generalize the work on Laplacian system solving to make the algorithms obtained via  ...  This iterative procedure can be viewed as a "box-constrained" variant of the Newton's method.  ... 
arXiv:1704.02310v2 fatcat:dcsy6vs65zgizoil5yp43mqfiy

Book announcements

1990 Discrete Applied Mathematics  
Quasi-Newton methods (Hessian matrix updates. The DFP and BFGS formulas). Finite differences and scaling (Finite differences. Scaling). Active set strategies (Adding and deleting constraints.  ...  Nature of stationary points). Convexity (Convex sets and functions. Differentiable functions). Local exploration (Gradient descent. Newton's method).  ... 
doi:10.1016/0166-218x(90)90013-3 fatcat:5jdut3iylrffvcnyiogm2tfe24

Parallel nonlinear predictive control

Anthony Kelman, Francesco Borrelli
2012 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton)  
We present a numerical study examining the capabilities of state-of-the-art interior point optimization solvers to utilize parallel linear algebra.  ...  We focus on the ability of these methods to handle, in a systematic way, large-scale systems with nonlinearities and constraints.  ...  Watson Research Center for solving an installation issue with WSMP, and the Ipopt authors for all of their development efforts.  ... 
doi:10.1109/allerton.2012.6483201 dblp:conf/allerton/KelmanB12 fatcat:mj3ygl6lwvhufh6i7llumnfomq

Least squares problems with inequality constraints as quadratic constraints

Jodi L. Mead, Rosemary A. Renaut
2010 Linear Algebra and its Applications  
Results are compared with solutions found by lsqlin, and the quadratically constrained formulation is solved using the L-curve, maximum a posteriori estimation (MAP), and the χ 2 regularization method.  ...  The χ 2 regularization method with quadratic constraints is the most effective method for solving least squares problems with box constraints. Linear least squares, Box constraints, Regularization  ...  Interior point methods use variants of Newton's method to solve the KKT equality conditions for (1) .  ... 
doi:10.1016/j.laa.2009.04.017 fatcat:bhwy5mtiubggtncia3tku5v5ru

Numerical methods for large-scale nonlinear optimization

Nick Gould, Dominique Orban, Philippe Toint
2005 Acta Numerica  
State-of-the-art algorithms for solving unconstrained, bound-constrained, linearly-constrained and nonlinearly-constrained problems are discussed.  ...  Recent developments in numerical methods for solving large differentiable nonlinear optimization problems are reviewed.  ...  Numerical methods for large-scale nonlinear optimization  ... 
doi:10.1017/s0962492904000248 fatcat:dz3twiovzfekxawl2h3wfi3m44

A computational framework for simulation of biogeochemical tracers in the ocean

Samar Khatiwala
2007 Global Biogeochemical Cycles  
This paper describes the key features and advantages of the transport matrix method, and illustrates its application to a series of realistic problems in chemical and biological oceanography.  ...  the task of simulating tracers to a sequence of simple matrix-vector products.  ...  I am grateful to Stephanie Dutkiewicz and Mick Follows for sharing their expertise of the MIT biogeochemical model, and Patrick Heimbach for computing the ECCO matrix.  ... 
doi:10.1029/2007gb002923 fatcat:d6eqk74dtjfaxfkzsfxt2r7mfu

Augmented Lagrangian and Proximal Methods for Constrained Structured Optimization

Alberto De Marchi, Matthias Gerdts
2021 Zenodo  
This is based on the proposed proximal augmented Lagrangian framework and weaves together the proximal point algorithm and a damped semismooth Newton's method.  ...  The inner subproblems can be solved by any method for structured optimization and the overall algorithm can be made matrix-free.  ...  Birgin and J. M. Martínez. "Large-scale active-set box-constrained optimization method with spectral projected gradients". In: Computational Optimization and Applications . ( ), pp. -. [ ] E. D.  ... 
doi:10.5281/zenodo.4972536 fatcat:2et67kr2b5bpfpnp5vodnr7eci

Optimal Experimental Design for Constrained Inverse Problems [article]

Lars Ruthotto, Julianne Chung, Matthias Chung
2017 arXiv   pre-print
To overcome the lack of non-differentiability in active set methods for inequality constraints problems, we use a relaxed interior point method.  ...  importance of OED for constrained problems.  ...  Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.  ... 
arXiv:1708.04740v1 fatcat:s7dtzjlg3jdhzp6hvlk65qstzu

Spectral Gradient Methods for Linearly Constrained Optimization

J. M. Martínez, E. A. Pilotta, M. Raydan
2005 Journal of Optimization Theory and Applications  
Each outer iteration involves the application of the previously defined preconditioned spectral gradient method for linear equality constrained problems.  ...  The associated extended KKT matrix remains constant unless the process is reinitiated. In ordinary inner iterations, only the right hand side of the KKT system changes.  ...  They have been applied to find local minimizers of large scale problems [4, 5, 22] , and also to explore faces of large dimensions in box-constrained optimization (see [3] and [12] ).  ... 
doi:10.1007/s10957-005-2093-3 fatcat:cg4vefgymvewtp2l3usuczbglu

Faster and more accurate computation of the H_∞ norm via optimization [article]

Peter Benner, Tim Mitchell
2019 arXiv   pre-print
Unlike the standard algorithm, our modified approach can also calculate the H_∞ norm to full precision with little extra work, and also offers more opportunity to further accelerate its performance via  ...  In this paper, we propose an improved method for computing the H_∞ norm of linear dynamical systems that results in a code that is often several times faster than existing methods.  ...  iterations and particularly when using the Newton's method variants.  ... 
arXiv:1707.02497v6 fatcat:q27l6n2zvrdwfbw2svxe7bqqni

Fast Parallel Algorithms for Short-Range Molecular Dynamics

Steve Plimpton
1995 Journal of Computational Physics  
, and Cray T3D.  ...  Trade-offs between the three algorithms and guidelines for adapting them to more complex molecular dynamics simulations are also discussed.  ...  They include particle-mesh algorithms [31] which scale as f (M )N where M is the number of mesh points, hierarchical methods [6] which scale as N log(N ), and fast-multipole methods [23] which scale  ... 
doi:10.1006/jcph.1995.1039 fatcat:2xmytcwf3ncifdw4lak3jaj6bq

Mathematical programs with complementarity constraints in traffic and telecommunications networks

D. Ralph
2008 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences  
has paved the way for finding local solutions of MPCCs via standard NLP techniques.  ...  An example is toll design in traffic networks, which attempts to reduce total travel time by choosing which arcs to toll and what toll levels to impose.  ...  The author gratefully acknowledges the input of Frank Kelly, Gaurav Raina, Mike Smith and Agachai Sumalee on the literature of network equilibria, and Michael Ferris, Jong-Shi Pang and Stefan Scholtes  ... 
doi:10.1098/rsta.2008.0026 pmid:18325868 fatcat:uphboppnineahftr7hvttac6iq

Two numerical methods for optimizing matrix stability

James V. Burke, Adrian S. Lewis, Michael L. Overton
2002 Linear Algebra and its Applications  
The spectral abscissa α(X) is a continuous but non-smooth, in fact non-Lipschitz, function of the matrix argument X, and finding a global minimizer of α(A(x)) is hard.  ...  Burke et al. / Linear Algebra and its Applications 351-352 (2002) approaching α(X) as δ → 0 and the largest eigenvalue of 1 2 (X + X T ) as δ → 1.  ...  The approach fell out of favor in the 1970s but was revived in the 1980s and 1990s [32] , following the explosion of interest in interior point methods for linear and convex programming.  ... 
doi:10.1016/s0024-3795(02)00260-4 fatcat:kor665cn3vbr5dhdmn5pcwaaoe

Differentiable Collision Avoidance Using Collision Primitives [article]

Simon Zimmermann, Matthias Busenhart, Simon Huber, Roi Poranne, Stelian Coros
2022 arXiv   pre-print
Optimization-based motion planning is one method, that often heavily relies on distance computations between robots and obstacles.  ...  To improve performance, many different methods suggested to use collision primitives, i.e. simple shapes that approximate the more complex rigid bodies, and that are simpler to compute distances to and  ...  Our method produced smoother velocities and found a more direct path. method and use it as a starting point for the optimization.  ... 
arXiv:2204.09352v1 fatcat:fhfrnxzwyngl7gmhm6exlsltxm

A Scaled Gradient Projection Method for Bayesian Learning in Dynamical Systems

S. Bonettini, A. Chiuso, M. Prato
2015 SIAM Journal on Scientific Computing  
In particular, we propose both a generalization of the split gradient approach to design the scaling matrix in the presence of box constraints, and an effective implementation of the gradient and objective  ...  In this paper we address this problem by means of a scaled gradient projection algorithm, in which the scaling matrix and the steplength parameter play a crucial role to provide a meaning solution in a  ...  To motivate the introduction of the scaling matrix, one can think, for example, of the Newton's method, which actually scales the gradient direction with the inverse Hessian, while other practical choices  ... 
doi:10.1137/140973529 fatcat:gliuual2mnfalgx34b3qzfrpia
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