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Direct Solution of Linear Systems of Size 109 Arising in Optimization with Interior Point Methods [chapter]

Jacek Gondzio, Andreas Grothey
2006 Lecture Notes in Computer Science  
Solution methods for very large scale optimization problems are addressed in this paper. Interior point methods are demonstrated to provide unequalled efficiency in this context.  ...  They need a small (and predictable) number of iterations to solve a problem. A single iteration of interior point method requires the solution of indefinite system of equations.  ...  Acknowledgements We are grateful to the EPCC for allowing us to use the BlueGene service and to Dr Joachim Hein in particular for his help in running OOPS on this machine.  ... 
doi:10.1007/11752578_62 fatcat:oj4sszgopfdexh3iurlfbvro2y

Page 5035 of Mathematical Reviews Vol. , Issue 2004f [page]

2004 Mathematical Reviews  
Note that other techniques that use iterative ap- proaches to solving the linear systems that arise in interior-point methods for LPs appear in, e.g., [M. G. C. Resende, K. G. Ra- makrishnan and Z.  ...  However, there are few successful iterative tech- niques in the literature for linear programming problems that compete with interior-point methods.  ... 

Recent Developments in Interior-Point Methods [chapter]

Stephen J. Wright
2000 IFIP Advances in Information and Communication Technology  
Interior-point methodology has been used as part of the solution strategy in many other optimization contexts as well, including analytic center methods and column-generation algorithms for large linear  ...  The modern era of interior-point methods dates to 1984, when Karmarkar proposed his algorithm for linear programming.  ...  Department of Energy, under Contract W-31-109-Eng-38.  ... 
doi:10.1007/978-0-387-35514-6_14 fatcat:chzlfupe7bbhbdyynhatlckjcu

A truncated primal-infeasible dual-feasible network interior point method

L. F. Portugal, M. G. C. Resende, G. Veiga, J. J. J�dice
2000 Networks  
for the solution of the system.  ...  In each iteration, the linear system that determines the search direction is computed inexactly, and the norm of the resulting residual vector is used in the stopping criteria of the iterative solver employed  ...  Suppose that we want to apply an iterative procedure to approximately solve the system of normal equations that arises in interior point methods.  ... 
doi:10.1002/(sici)1097-0037(200003)35:2<91::aid-net1>;2-t fatcat:jxouuat4gve6jj44uccabnq72q

Large scale portfolio optimization with piecewise linear transaction costs

Marina Potaptchik, Levent Tunçel, Henry Wolkowicz
2008 Optimization Methods and Software  
The problem is first solved approximately using a primal-dual interior-point method applied to the smoothed problem.  ...  The main departure from the classical quadratic programming formulation is the inclusion in the objective function of piecewise linear, separable functions representing the transaction costs.  ...  The specialized interior-point method ensures a major savings in total computation time. The active set method guarantees high accuracy of our final solution.  ... 
doi:10.1080/00207160802263858 fatcat:djlosobunvbfjmbb2ra4y263pa

Inertia-Revealing Preconditioning For Large-Scale Nonconvex Constrained Optimization

Olaf Schenk, Andreas Wächter, Martin Weiser
2009 SIAM Journal on Scientific Computing  
The suitability of the heuristics for application in optimization methods is verified on an interior point method applied to the CUTE and COPS test problems, on large-scale 3D PDE-constrained optimal control  ...  This condition can be verified using the signs of the KKT's eigenvalues (inertia), which are usually available from direct solvers for the arising linear saddle point problems.  ...  Then, we will evaluate the performance of the method within an interior-point optimization method on the popular CUTE [8] and COPS [7] The linear saddle-point systems (5) are solved by a preconditioned  ... 
doi:10.1137/070707233 fatcat:3c76a2awkzahbnhftlpkfdfsem

Parallel interior-point solver for structured linear programs

Jacek Gondzio, Robert Sarkissian
2003 Mathematical programming  
Issues of implementation of an object-oriented library for parallel interior-point methods are addressed. The solver can easily exploit any special structure of the underlying optimization problem.  ...  The efficiency of the solver is illustrated on several problems arising in the optimization of networks.  ...  Acknowledgements We are grateful to Ken McKinnon for his careful reading of the paper and his insightful comments, and to Andreas Grothey for help in running the parallel version of the solver.  ... 
doi:10.1007/s10107-003-0379-5 fatcat:6sj54wj6pfhaxmdfzbf62zw4bq

Application of Interior-Point Methods to Model Predictive Control

C. V. Rao, S. J. Wright, J. B. Rawlings
1998 Journal of Optimization Theory and Applications  
We present a structured interior-point method for the e cient solution of the optimal control problem in model predictive control MPC.  ...  The cost of this approach is linear in the horizon length, compared with cubic growth for a naive approach.  ...  Department of Energy, under Contract W-31-109-Eng-38, and a grant from Aspen Technology Inc.  ... 
doi:10.1023/a:1021711402723 fatcat:jdqqg3qcn5fd5nt6q3pzceojra

Interior Point Methods for Combinatorial Optimization [chapter]

John E. Mitchell, Panos M. Pardalos, Mauricio G. C. Resende
1998 Handbook of Combinatorial Optimization  
In this paper, we review recent interior point approaches for solving combinatorial optimization problems.  ...  We discuss in detail tecniques for linear and network programming, branch and bound and branch and cut methods, nonconvex potential function minimization, lower bounding techniques, and semidefinite programming  ...  The second author acknowledges support in part by NSF grant BIR-9505913 and U.S. Department of Air Force grant F08635-92-C-0032.  ... 
doi:10.1007/978-1-4613-0303-9_4 fatcat:jqlwni3hrbggze4w74hww2gsre

Interior Point Methods for Combinatorial Optimization [chapter]

John E. Mitchell
1996 Applied Optimization  
In this paper, we review recent interior point approaches for solving combinatorial optimization problems.  ...  We discuss in detail tecniques for linear and network programming, branch and bound and branch and cut methods, nonconvex potential function minimization, lower bounding techniques, and semidefinite programming  ...  The second author acknowledges support in part by NSF grant BIR-9505913 and U.S. Department of Air Force grant F08635-92-C-0032.  ... 
doi:10.1007/978-1-4613-3449-1_11 fatcat:5utlmdqf4fbxpbrlvxdmblrgde

A Survey of Recent Scalability Improvements for Semidefinite Programming with Applications in Machine Learning, Control, and Robotics [article]

Anirudha Majumdar, Georgina Hall, Amir Ali Ahmadi
2019 arXiv   pre-print
approximate solutions to semidefinite programs, (iii) more scalable algorithms that rely on augmented Lagrangian techniques and the alternating direction method of multipliers, and (iv) approaches that  ...  trade off scalability with conservatism (e.g., by approximating semidefinite programs with linear and second-order cone programs).  ...  This work is partially supported by the DARPA Young Faculty Award, the CAREER Award of the NSF, the Google Faculty Award, the Innovation Award of the School of Engineering and Applied Sciences at Princeton  ... 
arXiv:1908.05209v3 fatcat:g2vqfhv27vgddbv7l4xciywf4u

Simulation and optimization of dynamic flux balance analysis models using an interior point method reformulation

Felipe Scott, Pamela Wilson, Raúl Conejeros, Vassilios S. Vassiliadis
2018 Computers and Chemical Engineering  
By exploiting the first-order optimality conditions of the interior-point problem, and with further transformations, the approach results in a system of implicit ordinary differential equations.  ...  for the solution of LP problems, section 3 presents the interior 131 point based formulation to solve dFBA models as implicit systems of ODEs.  ...  However, we are solving non-linear systems which arise from complementarity conditions of an interior point reformulation of the LP, and such applications require a customized Newton solver which retracts  ... 
doi:10.1016/j.compchemeng.2018.08.041 fatcat:fgwydxbtencqfbse7nyc3g5jli

14th International Symposium on Mathematical Programming

1990 Mathematical programming  
to the well known serious steps und null steps of bundle methods a third type of steps is used to generate the model of dom .  ...  A v ariation of bundle methods is presented by means of which such a t ype of problems can be solved.In this variation a dynamic polyhedral model of dom is generated by means of normal vectors.In addition  ...  Keywords: interior point methods -numerical stability -pivoting -primal-dual methods We consider numerical stability problems arising in interior point methods for linear programming in primal degenerate  ... 
doi:10.1007/bf01580875 fatcat:3jtclwmntzgjxkqs5uecombdaa

Page 7062 of Mathematical Reviews Vol. , Issue 97K [page]

1997 Mathematical Reviews  
are of order O(n), where n is the number of mesh points in one direction.  ...  Summary: “We discuss the solution of Hermitian positive-definite systems Ax = b by the preconditioned conjugate gradient method with a preconditioner M.  ... 

Constraint interface preconditioning for topology optimization problems [article]

Michal Kocvara and Daniel Loghin and James Turner
2015 arXiv   pre-print
The discretization of constrained nonlinear optimization problems arising in the field of topology optimization yields algebraic systems which are challenging to solve in practice, due to pathological  ...  In this work we propose a methodology which brings together existing fast algorithms, namely, interior-point for the optimization problem and a novel substructuring domain decomposition method for the  ...  the Optimality Criteria method for topology optimization, which, unlike the interior point approach in this paper, requires the solution of a linear elasticity problem at each step. 7.  ... 
arXiv:1510.04568v1 fatcat:t2htymozvbhg7ccjrglixj6foy
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