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Solving Quadratic Programs to High Precision using Scaled Iterative Refinement [article]

Tobias Weber, Sebastian Sager, Ambros Gleixner
2018 arXiv   pre-print
We contribute on three levels to overcome this issue. First, we present a novel refinement algorithm to solve QPs to arbitrary precision.  ...  It iteratively solves refined QPs, assuming a floating-point QP solver oracle. We prove linear convergence of residuals and primal errors.  ...  Iterative Refinement for Quadratic Programming To solve quadratic programs to arbitrary, a priori specified precisions, we apply the refinement idea from the previous section iteratively in Algorithm 1  ... 
arXiv:1803.07178v1 fatcat:bou6l3gtnffzhkt6o7yph4zhia

Solving quadratic programs to high precision using scaled iterative refinement

Tobias Weber, Sebastian Sager, Ambros Gleixner
2019 Mathematical Programming Computation  
We contribute on three levels to overcome this issue. First, we present a novel refinement algorithm to solve QPs to arbitrary precision.  ...  It iteratively solves refined QPs, assuming a floating-point QP solver oracle. We prove linear convergence of residuals and primal errors.  ...  The iterative refinement algorithm for quadratic programming To solve quadratic programs to arbitrary, a priori specified precision, we apply the refinement idea from the previous section iteratively as  ... 
doi:10.1007/s12532-019-00154-6 fatcat:q2h2nstsjvdxhhs5lglfilfplm

Solving quadratic programs to high precision using scaled iterative refinement [article]

Tobias Weber, Sebastian Sager, Ambros M. Gleixner, Universitäts- Und Landesbibliothek Sachsen-Anhalt, Martin-Luther Universität
2020
We contribute on three levels to overcome this issue. First, we present a novel refinement algorithm to solve QPs to arbitrary precision.  ...  It iteratively solves refined QPs, assuming a floating-point QP solver oracle. We prove linear convergence of residuals and primal errors.  ...  The iterative refinement algorithm for quadratic programming To solve quadratic programs to arbitrary, a priori specified precision, we apply the refinement idea from the previous section iteratively as  ... 
doi:10.25673/35092 fatcat:int7jhag2vemlenjt47dgq2b3y

A Family of High-Performance Solvers for Linear Model Predictive Control

Gianluca Frison, Leo Emil Sokoler, John Bagterp Jørgensen
2014 IFAC Proceedings Volumes  
In Model Predictive Control (MPC), an optimization problem has to be solved at each sampling time, and this has traditionally limited the use of MPC to systems with slow dynamic.  ...  This strategy combines a Riccati-like solver with the use of high-performance computing techniques: in particular, in this paper we explore the performance boost given by the use of single precision computation  ...  quadratic program min z 1 2 z Hz + g z s.t.  ... 
doi:10.3182/20140824-6-za-1003.01302 fatcat:cyukgamwkfgqrpdnbgqcqhdsca

A generalized conjugate gradient algorithm for solving a class of quadratic programming problems

Dianne Prost O'Leary
1980 Linear Algebra and its Applications  
This quadratic programming problem often arises in a form such that the matrix A is large and has a nonrandom sparsity pattern.  ...  With this notation, (2) is equivalent to The values of variables xr will be kept fixed during the inner iteration, which will try to force all variables y, to be zero by solving A"x,= b,-A"+ A" is positive  ...  Thus far we have developed a finite algorithm to solve the quadratic programming problem with upper and lower bounds.  ... 
doi:10.1016/0024-3795(80)90173-1 fatcat:mhrpt3swxzg6zmjkyipw57svwm

Linear programming using limited-precision oracles

Ambros Gleixner, Daniel E. Steffy
2019 Mathematical programming  
In this paper we introduce the idea of a limited-precision LP oracle and study how such an oracle could be used within a larger framework to compute exact precision solutions to LPs.  ...  Since the elimination algorithm of Fourier and Motzkin, many different methods have been developed for solving linear programs.  ...  Acknowledgements The authors would like to thank the anonymous reviewers for their detailed study of the manuscript and their comments, which were of exceptionally high quality.  ... 
doi:10.1007/s10107-019-01444-6 fatcat:slkfzhzqjjh5dki6shcunmqmtu

Linear Programming Using Limited-Precision Oracles [chapter]

Ambros Gleixner, Daniel E. Steffy
2019 Lecture Notes in Computer Science  
Keywords Linear programming · oracle complexity · Diophantine approximation · exact solutions · symbolic computation · rational arithmetic · extended-precision arithmetic · iterative refinement Mathematics  ...  In this paper we introduce the idea of a limited-precision LP oracle and study how such an oracle could be used within a larger framework to compute exact precision solutions to LPs.  ...  Acknowledgements The authors would like to thank the anonymous reviewers for their detailed study of the manuscript and their comments, which were of exceptionally high quality.  ... 
doi:10.1007/978-3-030-17953-3_30 fatcat:aq7mgi3ymjdi3jbtfhcdqp6u2m

QPDAS: Dual Active Set Solver for Mixed Constraint Quadratic Programming [article]

Mattias Fält, Pontus Giselsson
2019 arXiv   pre-print
There are two main contributions; we present a new way of factorizing the linear systems, and show how iterative refinement can be used to achieve good accuracy and to solve both types of sub-problems  ...  We present a method for solving the general mixed constrained convex quadratic programming problem using an active set method on the dual problem.  ...  Iterative refinement can then be used to solve the projection problem directly by applying it to the equationḠx = 0.  ... 
arXiv:1911.12662v1 fatcat:kfcsrto77zhx7a5kudcdzwjlme

OSQP: An Operator Splitting Solver for Quadratic Programs [article]

Bartolomeo Stellato, Goran Banjac, Paul Goulart, Alberto Bemporad, Stephen Boyd
2019 arXiv   pre-print
In addition, our technique is the first operator splitting method for quadratic programs able to reliably detect primal and dual infeasible problems from the algorithm iterates.  ...  We present a general purpose solver for convex quadratic programs based on the alternating direction method of multipliers, employing a novel operator splitting technique that requires the solution of  ...  First order optimization methods for solving quadratic programs date to the 1950s [FW56] .  ... 
arXiv:1711.08013v3 fatcat:33o5es5nmndwncawsvsbfowzja

Linear algebra software on IBM and CRAY computers

J.-Fr. Hake, W. Homberg
1989 Journal of Computational and Applied Mathematics  
This approach can simplify code development and introduce know-how to the user's code.  ...  The acceptance of mathematical software libraries strongly depends on the quality of their components, because the library routines are often used as building blocks in more complex computer codes.  ...  Siegfried Knecht helped substantially to produce the storage access patterns of the subroutines LU, LUS and L2LRG. The authors also would like to thank the referee for his very helpful comments.  ... 
doi:10.1016/0377-0427(89)90303-8 fatcat:juwpi5cvj5a6xokfhtqybdjgoa

Downwash-Aware Trajectory Planning for Large Quadrotor Teams [article]

James A. Preiss, Wolfgang Hönig, Nora Ayanian, Gaurav S. Sukhatme
2017 arXiv   pre-print
We demonstrate the computational efficiency in simulation with up to 200 robots and the physical plausibility with an experiment with 32 nano-quadrotors.  ...  Our method decomposes the planning problem into two stages: a discrete planner operating on a graph representation of the workspace, and a continuous refinement that converts the non-smooth graph plan  ...  The perrobot trajectory optimization quadratic programs (14) are solved using Matlab's quadprog solver.  ... 
arXiv:1704.04852v2 fatcat:pj3stsw6hje45bs7qmtjqnx2py

Multiscale Strategies for Computing Optimal Transport [article]

Samuel Gerber, Mauro Maggioni
2017 arXiv   pre-print
The multiscale decomposition yields a sequence of optimal transport problems, that are solved in a top-to-bottom fashion from the coarsest to the finest scale.  ...  It is particularly well-suited for point sets that are in high-dimensions, but are close to being intrinsically low-dimensional.  ...  for paths and smaller linear programs have to be solved in each refinement step.  ... 
arXiv:1708.02469v1 fatcat:6m4xzllpdfdmflxgnoh4gf7jzu

Page 4323 of Mathematical Reviews Vol. , Issue 2000f [page]

2000 Mathematical Reviews  
The technique can be used for iterative refinement of solutions obtained by direct methods since it is particularly suited for high precision simultaneous ap- proximation of all the roots.  ...  , Alain Solving triangular algebraic systems by means of simultaneous iterations.  ... 

Implicit solvers for large-scale nonlinear problems

David E Keyes, Daniel R Reynolds, Carol S Woodward
2006 Journal of Physics, Conference Series  
generate an iterate that lies within the radius of quadratic convergence.  ...  Recent advances in iterative methods have made fully implicit formulations a viable option for solution of these large-scale problems.  ...  Acknowledgements The authors wish to thank Ravi Samtaney for use of his magnetic reconnection code to generate the Sweet-Parker scaling example in Section 5.  ... 
doi:10.1088/1742-6596/46/1/060 fatcat:lsgu3zpp65g3zorhv3jtcizrfm

Why does Astrée scale up?

Patrick Cousot, Radhia Cousot, Jérôme Feret, Laurent Mauborgne, Antoine Miné, Xavier Rival
2009 Formal methods in system design  
What makes ASTRÉE such an innovative tool is its scalability, while retaining the required precision, when it is used to analyze a specific class of programs: that of reactive control-command software.  ...  We discuss the way the precision of the semantics was tuned in ASTRÉE in order to scale up, the differences with some more academic approaches and some of the dead-ends we explored.  ...  That is a sound verification, precise enough to show the absence of errors and capable of scaling to their large critical programs.  ... 
doi:10.1007/s10703-009-0089-6 fatcat:lf5bl7rtznfjtlriqu62gvs35m
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