Preconditioning of conjugate gradient iterations in interior point MPC method

Abstract There are several efficient direct solvers for structured systems of linear equations defining search directions in primal-dual interior point methods applied to constrained model predictive control problems. We propose reusing matrix decompositions of direct solvers as preconditioners in Krylov-subspace methods applied to subsequent iterations of the interior point method, which results in at least halving its asymptotic computational complexity. We also analyze sensitivity of direct solvers to the regularization parameters.

[1]  M. Benzi,et al.  Some Preconditioning Techniques for Saddle Point Problems , 2008 .

[2]  Andrew Knyazev,et al.  Sparse preconditioning for model predictive control , 2015, 2016 American Control Conference (ACC).

[3]  E. Kerrigan,et al.  Preconditioners for inexact interior point methods for predictive control , 2010, Proceedings of the 2010 American Control Conference.

[4]  Eric C. Kerrigan,et al.  A Stable and Efficient Method for Solving a Convex Quadratic Program with Application to Optimal Control , 2012, SIAM J. Optim..

[5]  Eric C. Kerrigan,et al.  A fast well-conditioned interior point method for predictive control , 2010, 49th IEEE Conference on Decision and Control (CDC).

[6]  John Bagterp Jørgensen,et al.  High-performance small-scale solvers for linear Model Predictive Control , 2014, 2014 European Control Conference (ECC).

[7]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[8]  Andrew Knyazev,et al.  Preconditioned Continuation Model Predictive Control , 2015, SIAM Conf. on Control and its Applications.

[9]  Jacek Gondzio,et al.  Convergence Analysis of an Inexact Feasible Interior Point Method for Convex Quadratic Programming , 2012, SIAM J. Optim..

[10]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[11]  Stephen J. Wright.  Primal-Dual Interior-Point Methods , 1997, Other Titles in Applied Mathematics.

[12]  Johannes P. Schlöder,et al.  A factorization with update procedures for a KKT matrix arising in direct optimal control , 2011, Math. Program. Comput..

[13]  Robert J. Vanderbei,et al.  Symmetric Quasidefinite Matrices , 1995, SIAM J. Optim..

[14]  Margaret H. Wright,et al.  Ill-Conditioning and Computational Error in Interior Methods for Nonlinear Programming , 1998, SIAM J. Optim..

[15]  Chen Greif,et al.  A Preconditioner for Linear Systems Arising From Interior Point Optimization Methods , 2007, SIAM J. Sci. Comput..

[16]  John Bagterp Jørgensen,et al.  Efficient implementation of the Riccati recursion for solving linear-quadratic control problems , 2013, 2013 IEEE International Conference on Control Applications (CCA).

[17]  Stephen P. Boyd,et al.  Fast Model Predictive Control Using Online Optimization , 2010, IEEE Transactions on Control Systems Technology.

[18]  Gene H. Golub,et al.  Methods for modifying matrix factorizations , 1972, Milestones in Matrix Computation.

[19]  Hans Joachim Ferreau,et al.  Efficient Numerical Methods for Nonlinear MPC and Moving Horizon Estimation , 2009 .

[20]  Jacek Gondzio,et al.  Interior point methods 25 years later , 2012, Eur. J. Oper. Res..

[21]  Stephen J. Wright,et al.  Application of Interior-Point Methods to Model Predictive Control , 1998 .

[22]  Gene H. Golub,et al.  Matrix computations , 1983 .

[23]  Jack Dongarra,et al.  LAPACK Users' Guide, 3rd ed. , 1999 .

[24]  M. Saunders,et al.  SOLVING REGULARIZED LINEAR PROGRAMS USING BARRIER METHODS AND KKT SYSTEMS , 1996 .

[25]  Andrew V. Knyazev.  Observations on degenerate saddle point problems , 2007 .

[26]  L. Biegler,et al.  Active set vs. interior point strategies for model predictive control , 2000, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[27]  Stefano Di Cairano,et al.  Block Structured Preconditioning within an Active-Set Method for Real-Time Optimal Control , 2018, 2018 European Control Conference (ECC).

[28]  Stefano Di Cairano,et al.  A regularized Newton solver for linear model predictive control , 2018, 2018 European Control Conference (ECC).

[29]  Andrew Knyazev,et al.  Preconditioning for continuation model predictive control , 2015, ArXiv.

[30]  Sanjay Mehrotra,et al.  On the Implementation of a Primal-Dual Interior Point Method , 1992, SIAM J. Optim..

[31]  D. Sorensen,et al.  A new class of preconditioners for large-scale linear systems from interior point methods for linear programming , 2005 .

[32]  Manfred Morari,et al.  Efficient interior point methods for multistage problems arising in receding horizon control , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).