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

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Abstract

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.

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Notes

  1. We use the expression “descent direction” with respect to the barrier function.

References

  1. Agullo, E., Demmel, J., Dongarra, J., Hadri, B., Kurzak, J., Langou, J., Ltaief, H., Luszczek, P., Tomov, S.: Numerical linear algebra on emerging architectures: the PLASMA and MAGMA projects. J. Phys. Conf. Ser. 180, 012037 (2009)

    Article  Google Scholar 

  2. Amestoy, P.R., Guermouche, A., LExcellent, J.Y., Pralet, S.: Hybrid scheduling for the parallel solution of linear systems. Parallel Comput. 32(2), 136–156 (2006)

    Article  MathSciNet  Google Scholar 

  3. Arora, Nikhil, Biegler, Lorenz T.: A trust region SQP algorithm for equality constrained parameter estimation with simple parameter bounds. Comput. Optim. Appl. 28(1), 51–86 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Balay, S., Brown, J., Buschelman, K., Eijkhout, V., Gropp, W., Kaushik, D., Knepley, M., McInnes, L., Smith, B., Zhang, H.: PETSc Users Manual Revision 3.4 (2013)

  5. Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Biegler, L.T., Zavala, V.M.: Large-scale nonlinear programming using IPOPT: an integrating framework for enterprise-wide dynamic optimization. Comput. Chem. Eng. 33(3), 575–582 (2009)

    Article  Google Scholar 

  7. Biros, G., Ghattas, O.: Parallel Lagrange–Newton–Krylov–Schur methods for PDE-constrained optimization. Part I: The Krylov–Schur solver. SIAM J. Sci. Comput. 27(2), 687–713 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Borzì, A., Schulz, V.: Multigrid methods for PDE optimization. SIAM Rev. 51(2), 361–395 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bunch, J.R., Kaufman, L.: Some stable methods for calculating inertia and solving symmetric linear systems. Math. Comput. 31, 163–179 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  10. Byrd, R.H., Gilbert, J.C., Nocedal, J.: A trust-region method based on interior-point techniques for nonlinear programming. Math. Program. 89, 149–185 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Byrd, R.H., Curtis, F.E., Nocedal, J.: An inexact Newton method for nonconvex equality constrained optimization. Math. Program. 122(2), 273–299 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cervantes, A.M., Wächter, A., Tütüncü, R.H., Biegler, L.T.: A reduced space interior point strategy for optimization of differential algebraic systems. Comput. Chem. Eng. 24(1), 39–51 (2000)

    Article  Google Scholar 

  13. Chiang, N., Grothey, A.: Solving security constrained optimal power flow problems by a structure exploiting interior point method. Optim. Eng. 16, 49–71 (2012)

    Article  MathSciNet  Google Scholar 

  14. Chiang, N., Petra, C.G., Zavala, V.M.: Structured nonconvex optimization of large-scale energy systems using PIPS-NLP. In: Proceedings of the 18th Power Systems Computation Conference (PSCC). Wroclaw, Poland (2014)

  15. Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust Region Methods, vol. 1. SIAM, Philadelphia (2000)

    Book  MATH  Google Scholar 

  16. Costa, M.P., Fernandes, E.M.G.P.: Assessing the potential of interior point barrier filter line search methods: nonmonotone versus monotone approach. Optimization 60(10–11), 1251–1268 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Curtis, F.E., Nocedal, J., Wächter, A.: A matrix-free algorithm for equality constrained optimization problems with rank-deficient Jacobians. SIAM J. Optim. 20(3), 1224–1249 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Curtis, F.E., Schenk, O., Wächter, A.: An interior-point algorithm for large-scale nonlinear optimization with inexact step computations. SIAM J. Sci. Comput. 32(6), 3447–3475 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Duff, I.S.: Ma57—a code for the solution of sparse symmetric definite and indefinite systems. ACM Trans. Math. Softw. 30, 118–144 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Duff, I.S., Reid, J.K.: MA27—A Set of Fortran Subroutines for Solving Sparse Symmetric Sets of Linear Equations. UKAEA Atomic Energy Research Establishment (1982)

  22. Gondzio, J., Sarkissian, R.: Parallel interior-point solver for structured linear programs. Math. Program. 96(3), 561–584 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gould, N.I.M., Hribar, M.E., Nocedal, J.: On the solution of equality constrained quadratic programming problems arising in optimization. SIAM J. Sci. Comput. 23(4), 1376–1395 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gould, N.I.M., Lucidi, S., Roma, M., Toint, P.L.: Solving the trust-region subproblem using the Lanczos method. SIAM J. Optim. 9(2), 504–525 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gould, N.I.M.: On practical conditions for the existence and uniqueness of solutions to the general equality quadratic programming problem. Math. Program. 32(1), 90–99 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  26. Haverbeke, N., Diehl, M., De Moor, B.: A structure exploiting interior-point method for moving horizon estimation. In: Proceedings of the 48th IEEE Conference on Decision and Control, pp. 1273–1278. IEEE (2009)

  27. Haynsworth, E.V.: Determination of the inertia of a partitioned Hermitian matrix. Linear Algebra Appl. 1(1), 73–81 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  28. Heinkenschloss, M., Ridzal, D.: A matrix-free trust-region SQP method for equality constrained optimization. SIAM J. Optim. 24(3), 1507–1541 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Heroux, M.A., Willenbring, J.M.: Trilinos Users Guide. Citeseer (2003)

  30. Kawajiri, Y., Biegler, L.T.: Optimization strategies for simulated moving bed and powerfeed processes. AIChE J. 52(4), 1343–1350 (2006)

    Article  Google Scholar 

  31. Lubin, M., Petra, C.G., Anitescu, M.: The parallel solution of dense saddle-point linear systems arising in stochastic programming. Optim. Methods Softw. 27(4–5), 845–864 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lubin, M., Petra, C.G., Anitescu, M., Zavala, V.M.: Scalable stochastic optimization of complex energy systems. In: International Conference for High Performance Computing, Networking, Storage and Analysis (SC), 2011, pp. 1–10. IEEE (2011)

  33. Petra, C.G., Schenk, O., Lubin, M., Gaertner, K.: An augmented incomplete factorization approach for computing the Schur complement in stochastic optimization. SIAM J. Sci. Comput. 36(2), C139–C162 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. Rao, C.V., Wright, S.J., Rawlings, J.B.: Application of interior-point methods to model predictive control. J. Optim. Theory Appl. 99(3), 723–757 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. Schenk, O., Wächter, A., Hagemann, M.: Matching-based preprocessing algorithms to the solution of saddle-point problems in large-scale nonconvex interior-point optimization. Comput. Optim. Appl. 36, 321–341 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Schenk, O., Wächter, A., Weiser, M.: Inertia-revealing preconditioning for large-scale nonconvex constrained optimization. SIAM J. Sci. Comput. 31(2), 939–960 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Soler, M., Olivares, A., Staffetti, E.: Hybrid optimal control approach to commercial aircraft trajectory planning. J. Guid. Control Dyn. 33(3), 985–991 (2010)

    Article  Google Scholar 

  38. Wächter, A., Biegler, L.T.: On the implementation of a primal–dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Program. 106, 25–57 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wächter, A., Biegler, L.T.: Line search filter methods for nonlinear programming: motivation and global convergence. SIAM J. Optim. 16(1), 1–31 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Waltz, R.A., Morales, J.L., Nocedal, J., Orban, D.: An interior algorithm for nonlinear optimization that combines line search and trust region steps. Math. Program. 107(3), 391–408 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wang, Y., Boyd, S.: Fast model predictive control using online optimization. IEEE Trans. Control Syst. Technol. 18(2), 267–278 (2010)

    Article  Google Scholar 

  42. Zavala, V.M.: Stochastic optimal control model for natural gas networks. Comput. Chem. Eng. 64, 103–113 (2014)

    Article  Google Scholar 

  43. Zavala, V.M., Biegler, L.T.: Large-scale parameter estimation in low-density polyethylene tubular reactors. Ind. Eng. Chem. Res. 45(23), 7867–7881 (2006)

    Article  Google Scholar 

  44. Zavala, V.M., Laird, C.D., Biegler, L.T.: Interior-point decomposition approaches for parallel solution of large-scale nonlinear parameter estimation problems. Chem. Eng. Sci. 63(19), 4834–4845 (2008)

    Article  Google Scholar 

  45. Zenios, S., Lasken, R.: Nonlinear network optimization on a massively parallel connection machine. Ann. Oper. Res. 14(1), 147–165 (1988)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Contract No. DE-AC02-06CH11357. We thank Frank Curtis and Jorge Nocedal for technical discussions. Victor M. Zavala acknowledges funding from the DOE Office of Science under the Early Career program. We also acknowledge the computing resources provided by the Laboratory Computing Resource Center at Argonne National Laboratory.

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Authors and Affiliations

  1. Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL, 60439, USA

    Nai-Yuan Chiang

  2. Department of Chemical and Biological Engineering, University of Wisconsin-Madison, 1415 Engineering Hall, Madison, WI, 53706, USA

    Victor M. Zavala

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  1. Nai-Yuan Chiang
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  2. Victor M. Zavala
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Correspondence to Victor M. Zavala.

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Chiang, NY., Zavala, V.M. An inertia-free filter line-search algorithm for large-scale nonlinear programming. Comput Optim Appl 64, 327–354 (2016). https://doi.org/10.1007/s10589-015-9820-y

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