An inertia-free filter line-search algorithm for large-scale nonlinear programming

We present a filter line-search algorithm that does not require inertia information of the linear system. This feature enables the use of a wide range of linear algebra strategies and libraries, which is essential to tackle large-scale problems on modern computing architectures. The proposed approach performs curvature tests along the search step to detect negative curvature and to trigger convexification. We prove that the approach is globally convergent and we implement the approach within a parallel interior-point framework to solve large-scale and highly nonlinear problems. Our numerical tests demonstrate that the inertia-free approach is as efficient as inertia detection via symmetric indefinite factorizations. We also demonstrate that the inertia-free approach can lead to reductions in solution time because it reduces the amount of convexification needed.

[1]  Matthew G. Knepley,et al.  PETSc Users Manual (Rev. 3.4) , 2014 .

[2]  L. Biegler,et al.  Large-Scale Parameter Estimation in Low-Density Polyethylene Tubular Reactors , 2006 .

[3]  Nicholas I. M. Gould,et al.  Solving the Trust-Region Subproblem using the Lanczos Method , 1999, SIAM J. Optim..

[4]  L. Biegler,et al.  A reduced space interior point strategy for optimization of differential algebraic systems , 2000 .

[5]  Michael A. Heroux,et al.  Trilinos users guide. , 2003 .

[6]  Nicholas I. M. Gould,et al.  Trust Region Methods , 2000, MOS-SIAM Series on Optimization.

[7]  Iain S. Duff,et al.  MA57---a code for the solution of sparse symmetric definite and indefinite systems , 2004, TOMS.

[8]  Gene H. Golub,et al.  Numerical solution of saddle point problems , 2005, Acta Numerica.

[9]  Alberto Olivares,et al.  Engineering Notes Hybrid Optimal Control Approach to Commercial Aircraft Trajectory Planning , 2010 .

[10]  Olaf Schenk,et al.  Inertia-Revealing Preconditioning For Large-Scale Nonconvex Constrained Optimization , 2008, SIAM J. Sci. Comput..

[11]  Cosmin G. Petra,et al.  An Augmented Incomplete Factorization Approach for Computing the Schur Complement in Stochastic Optimization , 2014, SIAM J. Sci. Comput..

[12]  Nicholas I. M. Gould,et al.  On practical conditions for the existence and uniqueness of solutions to the general equality quadratic programming problem , 1985, Math. Program..

[13]  Patrick Amestoy,et al.  Hybrid scheduling for the parallel solution of linear systems , 2006, Parallel Comput..

[14]  Jorge Nocedal,et al.  A trust region method based on interior point techniques for nonlinear programming , 2000, Math. Program..

[15]  Lorenz T. Biegler,et al.  On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming , 2006, Math. Program..

[16]  Victor M. Zavala,et al.  Interior-point decomposition approaches for parallel solution of large-scale nonlinear parameter estimation problems , 2008 .

[17]  Victor M. Zavala,et al.  Stability of multiobjective predictive control: A utopia-tracking approach , 2012, Autom..

[18]  Matthias Heinkenschloss,et al.  A Matrix-Free Trust-Region SQP Method for Equality Constrained Optimization , 2014, SIAM J. Optim..

[19]  Luca Bergamaschi,et al.  Erratum to: Inexact constraint preconditioners for linear systems arising in interior point methods , 2011, Comput. Optim. Appl..

[20]  L. Biegler,et al.  Optimization strategies for simulated moving bed and PowerFeed processes , 2006 .

[21]  Jorge Nocedal,et al.  A Matrix-Free Algorithm for Equality Constrained Optimization Problems with Rank-Deficient Jacobians , 2009, SIAM J. Optim..

[22]  Victor M. Zavala,et al.  Large-scale nonlinear programming using IPOPT: An integrating framework for enterprise-wide dynamic optimization , 2009, Comput. Chem. Eng..

[23]  Lorenz T. Biegler,et al.  A Trust Region SQP Algorithm for Equality Constrained Parameter Estimation with Simple Parameter Bounds , 2004, Comput. Optim. Appl..

[24]  Victor M. Zavala,et al.  Scalable stochastic optimization of complex energy systems , 2011, 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC).

[25]  A. Grothey,et al.  Solving security constrained optimal power flow problems by a structure exploiting interior point method , 2015 .

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

[27]  Iain S. Duff,et al.  MA27 -- A set of Fortran subroutines for solving sparse symmetric sets of linear equations , 1982 .

[28]  Edite Manuela da G. P. Fernandes,et al.  Assessing the potential of interior point barrier filter line search methods: nonmonotone versus monotone approach , 2011 .

[29]  Nicholas I. M. Gould,et al.  On the Solution of Equality Constrained Quadratic Programming Problems Arising in Optimization , 2001, SIAM J. Sci. Comput..

[30]  Victor M. Zavala,et al.  Structured nonconvex optimization of large-scale energy systems using PIPS-NLP , 2014, 2014 Power Systems Computation Conference.

[31]  Lorenz T. Biegler,et al.  Line Search Filter Methods for Nonlinear Programming: Motivation and Global Convergence , 2005, SIAM J. Optim..

[32]  Olaf Schenk,et al.  An Interior-Point Algorithm for Large-Scale Nonlinear Optimization with Inexact Step Computations , 2010, SIAM J. Sci. Comput..

[33]  Stavros A. Zenios,et al.  Nonlinear network optimization on a massively parallel connection machine , 1988 .

[34]  William Gropp,et al.  PETSc Users Manual Revision 3.4 , 2016 .

[35]  Olaf Schenk,et al.  Matching-based preprocessing algorithms to the solution of saddle-point problems in large-scale nonconvex interior-point optimization , 2007, Comput. Optim. Appl..

[36]  Bart De Moor,et al.  A structure exploiting interior-point method for moving horizon estimation , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[37]  Mihai Anitescu,et al.  The parallel solution of dense saddle-point linear systems arising in stochastic programming , 2012, Optim. Methods Softw..

[38]  Jack Dongarra,et al.  Numerical linear algebra on emerging architectures: The PLASMA and MAGMA projects , 2009 .

[39]  E. Haynsworth.  Determination of the inertia of a partitioned Hermitian matrix , 1968 .

[40]  Jorge Nocedal,et al.  An interior algorithm for nonlinear optimization that combines line search and trust region steps , 2006, Math. Program..

[41]  Jorge J. Moré,et al.  Digital Object Identifier (DOI) 10.1007/s101070100263 , 2001 .

[42]  V. Zavala.  Real-Time Optimization Strategies for Building Systems† , 2013 .

[43]  J. Bunch,et al.  Some stable methods for calculating inertia and solving symmetric linear systems , 1977 .

[44]  Jorge Nocedal,et al.  Adaptive Barrier Update Strategies for Nonlinear Interior Methods , 2008, SIAM J. Optim..

[45]  Jacek Gondzio,et al.  Parallel interior-point solver for structured linear programs , 2003, Math. Program..

[46]  Alfio Borzì,et al.  Multigrid Methods for PDE Optimization , 2009, SIAM Rev..

[47]  Jorge Nocedal,et al.  An inexact Newton method for nonconvex equality constrained optimization , 2009, Math. Program..

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

[49]  George Biros,et al.  Parallel Lagrange-Newton-Krylov-Schur Methods for PDE-Constrained Optimization. Part I: The Krylov-Schur Solver , 2005, SIAM J. Sci. Comput..

[50]  Victor M. Zavala,et al.  Stochastic optimal control model for natural gas networks , 2014, Comput. Chem. Eng..