Kybernetika 55 no. 1, 183-202, 2019

A nonmonotone line search for the LBFGS method in parabolic optimal control problems

Omid Solaymani Fard, Farhad Sarani, Akbar Hashemi Borzabadi and Hadi NosratipourDOI: 10.14736/kyb-2019-1-0183

Abstract:

In this paper a nonmonotone limited memory BFGS (NLBFGS) method is applied for approximately solving optimal control problems (OCPs) governed by one-dimensional parabolic partial differential equations. A discretized optimal control problem is obtained by using piecewise linear finite element and well-known backward Euler methods. Afterwards, regarding the implicit function theorem, the optimal control problem is transformed into an unconstrained nonlinear optimization problem (UNOP). Finally the obtained UNOP is solved by utilizing the NLBFGS method. In comparison to other existing methods, the NLBFGS method shows a significant improvement especially for nonlinear and ill-posed control problems.

Keywords:

optimal control, parabolic partial differential equations, backward Euler method, nonmonotone LBFGS method

Classification:

65K10, 90C30, 90C53

paper.pdf

References:

  1. H. Albrecher, W. J. Runggaldier and W. Schachermayer: Advanced Financial Modelling. Radon series on computational and applied mathematics, Walter de Gruyter, 2009.   DOI:10.1515/9783110213140
  2. K. Amini, M. Ahookhosh and H. Nosratipour: An inexact line search approach using modified nonmonotone strategy for unconstrained optimization. Numer. Algor. 66 (2014), 49-78.   DOI:10.1007/s11075-013-9723-x
  3. S. Aniţa, V. Arn\u{a}utu and V. Capasso: An Introduction to Optimal Control Problems in Life Sciences and Economics: From Mathematical Models to Numerical Simulation with MATLAB. Birkhäuser, Boston 2011.   CrossRef
  4. M. S. Bazaraa, H. D. Sherali and C. M. Shetty: Nonlinear Programming: Theory and Algorithms. Wiley, New York 2006.   DOI:10.1002/0471787779
  5. A. Borzi and V. Schulz: Computational Optimization of Systems Governed by Partial Differential Equations. SIAM, 2012.   DOI:10.1137/1.9781611972054
  6. F. Brezzi and M. Fortin: Mixed and Hybrid Finite Element Methods. Springer, New York 2012.   DOI:10.1007/978-1-4612-3172-1
  7. S. Cantrell, C. Cosner and S. Ruan: Spatial Ecology. CRC Mathematical and Computational Biology, CRC Press 2009.   DOI:10.1201/9781420059861
  8. R. Y. Chang and S. Y. Yang: Solution of two point boundary value problems by generalized orthogonal polynomials and application to optimal control of lumped and distributed parameter systems. International Journal of Control 43 (1986), 1785-1802.   DOI:10.1080/00207178608933572
  9. P. Christofides, A. Armaou, Y. Lou and A. Varshney: Control and Optimization of Multiscale Process Systems, Control Engineering. Birkhäuser, Boston 2008.   DOI:10.1007/978-0-8176-4793-3
  10. E. De Klerk, C. Roos and T. Terlaky: Nonlinear Optimization. University Of Waterloo, Waterloo 2005.   CrossRef
  11. I. Griva, S. G. Nash and A. Sofer: Linear and Nonlinear Optimization. SIAM, Philadelphia 2009.   DOI:10.1137/1.9780898717730
  12. J. Haslinger and P. Neittaanmäki: Finite Element Approximation for Optimal Shape, Material and Topology Design. Wiley, 1996.   CrossRef
  13. M. Heinkenschloss: Numerical Solution of Implicitly Constrained Optimization Problems. CAAM Technical Report TR08-05, Rice University (2008).   CrossRef
  14. M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich: Optimization with PDE Constraints. Springer, Netherlands 2008.   CrossRef
  15. I. R. Horng and J. H. Chou: Application of shifted Chebyshev series to the optimal control of linear distributed-parameter systems. Int. J. Control 42 (1985), 233-241.   DOI:10.1080/00207178508933359
  16. W. W. Hu: Approximation and Control of the Boussinesq Equations with Application to Control of Energy Efficient Building Systems. Ph.D. Thesis, Department of Mathematics, Virginia Tech. 2012.   CrossRef
  17. Y. Ji, Y. Li, K. Zhang and X. Zhan: A new nonmonotone trust-region method of conic model for solving unconstrained optimization. J. Comput. Appl. Math. 233 (2010), 1746-1754.   DOI:10.1016/j.cam.2009.09.011
  18. K. Kunisch and S. Volkwein: Control of the burgers equation by a reduced-order approach using proper orthogonal decomposition. J. Optim. Theory Appl. 102 (1999), 345-371.   DOI:10.1023/a:1021732508059
  19. J. L. Lions: Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, 1971.   DOI:10.1007/978-3-642-65024-6
  20. D. Liu and J. Nocedal: On the limited memory BFGS method for large-scale optimization. Math. Program. 45 (1989), 503-528.   DOI:10.1007/bf01589116
  21. P. Merino: Finite element error estimates for an optimal control problem governed by the Burgers equation. Comput. Optim. Appl. 63 (2016), 793-824.   DOI:10.1007/s10589-015-9790-0
  22. C. Meyer, P. Philip and F. Tröltzsch: Optimal control of a semilinear PDE with nonlocal radiation interface conditions. SIAM J. Control Optim. 45 (2006), 699-721.   DOI:10.1137/040617753
  23. B. R. Noack, M. Morzynski and G. Tadmor: Reduced-Order Modelling for Flow Control. Springer, Vienna 2011.   DOI:10.1007/978-3-7091-0758-4
  24. J. Nocedal: Updating quasi-Newton matrices with limited storage. Math. Comput. 35 (1980) 773-782.   DOI:10.1090/s0025-5718-1980-0572855-7
  25. J. Nocedal and S. Wright: Numerical Optimization. Springer, New York 2006.   DOI:10.1007/b98874
  26. H. Nosratipour, A. H. Borzabadi and O. S. Fard: Optimal control of viscous Burgers equation via an adaptive nonmonotone Barzilai-Borwein gradient method. Int. J. Comput. Math. 95 (2018) 1858-1873.   DOI:10.1080/00207160.2017.1343472
  27. H. Nosratipour, A. H. Borzabadi and O. S. Fard: On the nonmonotonicity degree of nonmonotone line searches. Calcolo 54 (2017) 1217-1242.   DOI:10.1007/s10092-017-0226-3
  28. H. Nosratipour, O. S. Fard and A. H. Borzabadi: An adaptive nonmonotone global Barzilai-Borwein gradient method for unconstrained optimization. Optimization 66 (2017) 641-655.   DOI:10.1080/02331934.2017.1287702
  29. J. A. Rad, S. Kazem and K. Parand: Optimal control of a parabolic distributed parameter system via radial basis functions. Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 2559-2567.   DOI:10.1016/j.cnsns.2013.01.007
  30. M. Razzaghi and A. Arabshahi: Optimal control of linear distributed-parameter systems via polynomial series. Int. J. Systems Sci. 20 (1989), 1141-1148.   DOI:10.1080/00207728908910200
  31. Z. Sabeh, M. Shamsi and M. Dehghan: Distributed optimal control of the viscous Burgers equation via a Legendre pseudo-spectral approach. Math. Methods Appl. Sci. 39 (2016), 3350-3360.   DOI:10.1002/mma.3779
  32. G. Strang and G. Fix: An Analysis of the Finite Element Method. Wellesley-Cambridge Press, 2008.   CrossRef
  33. F. Tröltzsch: Optimal Control of Partial Differential Equations: Theory, Methods, and Applications. Graduate studies in mathematics, American Mathematical Society, 2010.   DOI:10.1090/gsm/112
  34. F. Tröltzsch and S. Volkwein: The SQP method for control constrained optimal control of the Burgers equation. ESAIM: COCV 6 (2001), 649-674.   DOI:10.1051/cocv:2001127
  35. F. S. Wang and J. B. Jian: A new nonmonotone linesearch SQP algorithm for unconstrained minimax problem. Numer. Funct. Anal. Optim. 35 (2014), 487-508.   DOI:10.1080/01630563.2013.873454
  36. F. Yılmaz and B. Karasözen: Solving distributed optimal control problems for the unsteady Burgers equation in COMSOL multiphysics. J. Comput. Appl. Math. 235 (2011), 4839-4850.   DOI:10.1016/j.cam.2011.01.002
  37. H. Zhang and W. W. Hager: A nonmonotone line search technique and its application to unconstrained optimization. SIAM J. Optim. 14 (2004), 1043-1056.   DOI:10.1137/s1052623403428208