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HyKKT: a hybrid direct-iterative method for solving KKT linear systems

Journal Article · · Optimization Methods and Software
 [1];  [2];  [1];  [2];  [1];  [3];  [4]
  1. Stanford Univ., CA (United States)
  2. Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
  3. Pacific Northwest National Laboratory (PNNL), Richland, WA (United States)
  4. Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
+ Show Author Affiliations

Here, we propose a solution strategy for the large indefinite linear systems arising in interior methods for nonlinear optimization. The method is suitable for implementation on hardware accelerators such as graphical processing units (GPUs). The current gold standard for sparse indefinite systems is the LBLT factorization where L is a lower triangular matrix and B is 1×1 or 2×2 block diagonal. However, this requires pivoting, which substantially increases communication cost and degrades performance on GPUs. Our approach solves a large indefinite system by solving multiple smaller positive definite systems, using an iterative solver on the Schur complement and an inner direct solve (via Cholesky factorization) within each iteration. Cholesky is stable without pivoting, thereby reducing communication and allowing reuse of the symbolic factorization. We demonstrate the practicality of our approach on large optimal power flow problems and show that it can efficiently utilize GPUs and outperform LBLT factorization of the full system.

Research Organization:
Pacific Northwest National Laboratory (PNNL), Richland, WA (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC)
Grant/Contract Number:
AC05-76RL01830
OSTI ID:
2446704
Report Number(s):
PNNL-SA--166808
Journal Information:
Optimization Methods and Software, Journal Name: Optimization Methods and Software Journal Issue: 2 Vol. 38; ISSN 1055-6788
Publisher:
Taylor & FrancisCopyright Statement
Country of Publication:
United States
Language:
English

References (34)

A comparative study of null-space factorizations for sparse symmetric saddle point systems: A comparative study of null-space factorizations journal June 2017
Knitro: An Integrated Package for Nonlinear Optimization book January 2006
On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming journal April 2005
An inertia-free filter line-search algorithm for large-scale nonlinear programming journal February 2016
HOPDM (version 2.12) — A fast LP solver based on a primal-dual interior point method journal August 1995
PaStiX: a high-performance parallel direct solver for sparse symmetric positive definite systems journal February 2002
Optimal control for a tuberculosis model with reinfection and post-exposure interventions journal August 2013
Accelerating sparse Cholesky factorization on GPUs journal November 2016
Block preconditioning for saddle point systems with indefinite (1, 1) block journal August 2007
A computational study of the use of an optimization-based method for simulating large multibody systems† journal December 2009
LOQO:an interior point code for quadratic programming journal January 1999
A statistical approach for inferring the 3D structure of the genome journal June 2014
Basker: A Threaded Sparse LU Factorization Utilizing Hierarchical Parallelism and Data Layouts
  • Booth, Joshua Dennis; Rajamanickam, Sivasankaran; Thornquist, Heidi
  • 2016 IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW) https://doi.org/10.1109/IPDPSW.2016.92
conference May 2016
Contact-Implicit Trajectory Optimization for Dynamic Object Manipulation conference November 2019
Non-Gaussian Chance-Constrained Trajectory Planning for Autonomous Vehicles Under Agent Uncertainty journal October 2020
Security Constrained Optimal Power Flow via proximal message passing conference March 2014
Stochastic Model Predictive Control for Building HVAC Systems: Complexity and Conservatism journal January 2015
Toward the Next Generation of Multiperiod Optimal Power Flow Solvers journal July 2018
A Survey of Distributed Optimization and Control Algorithms for Electric Power Systems journal November 2017
GPU-Accelerated Parallel Sparse LU Factorization Method for Fast Circuit Analysis journal March 2016
Implicit-Factorization Preconditioning and Iterative Solvers for Regularized Saddle-Point Systems journal January 2006
Solution of Sparse Indefinite Systems of Linear Equations journal September 1975
Line Search Filter Methods for Nonlinear Programming: Motivation and Global Convergence journal January 2005
On Solving Block-Structured Indefinite Linear Systems journal January 2003
Semilocal momentum-space regularized chiral two-nucleon potentials up to fifth order journal May 2018
An overview of SuperLU: Algorithms, implementation, and user interface journal September 2005
Algorithm 887: CHOLMOD, Supernodal Sparse Cholesky Factorization and Update/Downdate journal October 2008
Algorithm 907: KLU, A Direct Sparse Solver for Circuit Simulation Problems journal September 2010
A Distributed-Memory Package for Dense Hierarchically Semi-Separable Matrix Computations Using Randomization journal June 2016
MA57---a code for the solution of sparse symmetric definite and indefinite systems journal June 2004
Model predictive air path control for a two-stage turbocharged spark-ignition engine with low pressure exhaust gas recirculation journal July 2020
Computational approaches for RNA energy parameter estimation journal October 2010
Matrix Computations book February 2013
Methods of conjugate gradients for solving linear systems journal December 1952

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Related Subjects

97 MATHEMATICS AND COMPUTING
GPU
KKT systems
Optimization
interior methods
sparse matrix factorization