An Efficient Nonmonotone Method for State-Constrained Elliptic Optimal Control Problems

Abstract

This paper presents a novel numerical strategy based on combination of an adaptive semismooth Newton (ASN) method and the Lavrentiev regularization technique for the solution of elliptic optimal control problems with state constraints. Using the global convergence proof for a nonmonotone semismooth Newton method, we will exploit an adaptive nonmonotone line search method such that the nonmonotonicity degree of this method can be increased when the results are far from the optimum solution and it can be reduced when they are close to the optimizer. In this strategy, the role of the Lavrentiev regularization technique is converting the original optimal control problem to a regularized optimal control problem. Using the finite difference discretization scheme and a Newton–Cotes rule, the regularized optimal control problem is converted to a bound constrained optimization problem (BCOP). Then the ASN method is implemented to solve the resulting BCOP. Numerical results show the efficiency of the proposed procedure.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

References

  1. 1.

    Ahookhosh, M., Ghaderi, S.: On efficiency of nonmonotone Armijo-type line searches. Appl. Math. Model. 43, 170–190 (2017)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Amini, K., Ahookhosh, M., Nosratipour, H.: An inexact line search approach using modified nonmonotone strategy for unconstrained optimization. Numer. Algorithms 66, 49–78 (2014)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Anita, S., Arnautu, V., Capasso, V.: An Introduction to Optimal Control Problems in Life Sciences and Economics: From Mathematical Models to Numerical Simulation with MATLAB. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston (2011)

    MATH  Google Scholar 

  4. 4.

    Arqub, O.A.: The reproducing kernel algorithm for handling differential algebraic systems of ordinary differential equations. Math. Methods Appl. Sci. 39, 4549–4562 (2016a)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Arqub, O.A.: Approximate solutions of DASs with nonclassical boundary conditions using novel reproducing kernel algorithm. Fundam. Inform. 146, 231–254 (2016b)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Arqub, O.A., Shawagfeh, N.: Solving optimal control problems of Fredholm constraint optimality via the reproducing kernel Hilbert space method with error estimates and convergence analysis. Math. Methods Appl Sci. (2019). https://doi.org/10.1002/mma.5530

    Article  Google Scholar 

  7. 7.

    Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms. Wiley, New York (2006)

    MATH  Google Scholar 

  8. 8.

    Bergounioux, M., Kunisch, K.: Primal-dual strategy for state-constrained optimal control problems. Comput. Optim. Appl. 22, 193–224 (2002)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Bonettini, S.: A nonmonotone inexact Newton method. Optim. Methods Softw. 20, 475–491 (2005)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Borzì, A.: Smoothers for control- and state-constrained optimal control problems. Comput. Vis. Sci. 11, 59–66 (2008)

    MathSciNet  Google Scholar 

  11. 11.

    Borzì, A., Schulz, V.: Computational Optimization of Systems Governed by Partial Differential Equations. Computational Science and Engineering. SIAM, Philadelphia (2012)

    MATH  Google Scholar 

  12. 12.

    Cantrell, S., Cosner, C., Ruan, S.: Spatial Ecology, CRC Mathematical and Computational Biology. CRC Press, Boca Raton (2009)

    Google Scholar 

  13. 13.

    Capasso, V., Burkard, R., Deuflhard, P., Engl, H.W., Jameson, A., Periaux, J., Lions, J.L., Strang, G.: Computational Mathematics Driven by Industrial Problems. Lecture Notes in Mathematics. Springer, Berlin (2000)

    Google Scholar 

  14. 14.

    Chen, X., Nashed, Z., Qi, L.: Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM J. Numer. Anal. 38, 1200–1216 (2000)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Cherednichenko, S., Rösch, A.: Error estimates for the regularization of optimal control problems with pointwise control and state constraints. J. Anal. Appl. 27, 195–212 (2008)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Chi, X., Wei, H., Wan, Z., Zhu, Z.: Smoothing Newton algorithm for the circular cone programming with a nonmonotone line search. Acta Math. Sci. 37, 1262–1280 (2017)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Christofides, P., Armaou, A., Lou, Y., Varshney, A.: Control and Optimization of Multiscale Process Systems, Control Engineering. Birkhäuser, Boston (2008)

    MATH  Google Scholar 

  18. 18.

    Diehl, M., Glineur, F., Jarlebring, E., Michiels, W.: Recent Advances in Optimization and its Applications in Engineering. Springer, Berlin (2010)

    Google Scholar 

  19. 19.

    Fard, O.S., Borzabadi, A.H., Sarani, F.: An adaptive semismooth Newton method for approximately solving control-constrained elliptic optimal control problems. Trans. Inst. Meas. Control 41, 3010–3020 (2019)

    Google Scholar 

  20. 20.

    Field, D.A., Komkov, V.: Theoretical aspects of industrial design. In: Proceedings in Applied Mathematics Series. SIAM (1992)

  21. 21.

    Griva, I., Nash, S.G., Sofer, A.: Linear and Nonlinear Optimization, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2009)

    MATH  Google Scholar 

  22. 22.

    Hinze, M., Pinnau, R., Ulbrich, R., Ulbrich, S.: Optimization with PDE Constraints. Springer, Berlin (2008)

    MATH  Google Scholar 

  23. 23.

    Hu, W.W.: 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)

  24. 24.

    Ito, K., Kunisch, K.: Lagrange Multiplier Approach to Variational Problems and Applications. Society for Industrial and Applied Mathematics, Philadelphia (2008)

    MATH  Google Scholar 

  25. 25.

    Kimiaei, M., Rahpeymaii, F.: A new nonmonotone linesearch trust-region approach for nonlinear systems. TOP 27, 199–232 (2019)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Kimiaei, M.: A new class of nonmonotone adaptive trust region methods for nonlinear equations with box constraints. Calcolo 54, 769–812 (2017)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Kröner, A., Kunisch, K., Vexler, B.: Semismooth Newton methods for optimal control of the wave equation with control constraints. SIAM J. Control Optim. 49, 830–858 (2011)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, New York (1971)

    MATH  Google Scholar 

  29. 29.

    Luenberger, D.G., Ye, Y.: Linear and Nonlinear Programming. Springer, New York (2008)

    MATH  Google Scholar 

  30. 30.

    Manchanda, P., Lozi, R., Siddiqi, A.H.: Industrial Mathematics and Complex Systems: Emerging Mathematical Models. Methods and Algorithms, Industrial and Applied Mathematics. Springer, Singapore (2017)

    MATH  Google Scholar 

  31. 31.

    Meyer, C., Philip, P., Tröltzsch, F.: Optimal control of a semilinear PDE with nonlocal radiation interface conditions. SIAM J. Control Optim. 45, 699–721 (2006)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Meyer, C., Rösch, A., Tröltzsch, F.: Optimal control of PDEs with regularized pointwise state constraints. Comput. Optim. Appl. 33, 209–228 (2006)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Meyer, C., Tröltzsch, F.: On an elliptic optimal control problem with pointwise mixed control-state constraints, recent advances in optimization. Lect. Notes Econ. Math. Syst. 563, 187–204 (2006)

    MATH  Google Scholar 

  34. 34.

    Nocedal, J., Wright, S.: Numerical Optimization. Springer Series in Operations Research and Financial Engineering. Springer, New York (2006)

    Google Scholar 

  35. 35.

    Nosratipour, H., Borzabadi, A.H., Fard, O.S.: Optimal control of viscous Burgers equation via an adaptive nonmonotone Barzilai-Borwein gradient method. Int. J. Comput. Math. (2017). https://doi.org/10.1080/00207160.2017.1343472

    Article  MATH  Google Scholar 

  36. 36.

    Nosratipour, H., Borzabadi, A.H., Fard, O.S.: On the nonmonotonicity degree of nonmonotone line searches. Calcolo 54, 1217–1242 (2017b)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Nosratipour, H., Fard, O.S., Borzabadi, A.H.: An adaptive nonmonotone global Barzilai–Borwein gradient method for unconstrained optimization. Optimization 66, 641–655 (2017c)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Nosratipour, H., Fard, O.S., Borzabadi, A.H., Sarani, F.: Stable equilibrium configuration of two bar truss by an efficient nonmonotone global Barzilai–Borwein gradient method in a fuzzy environment. Afrika Matematika 28, 333–356 (2017d)

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Pang, J.S., Qi, L.: Nonsmooth equations: motivation and algorithms. SIAM J. Optim. 3, 443–465 (1993)

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993)

    MathSciNet  MATH  Google Scholar 

  41. 41.

    Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Shen, C., Leyffer, S., Fletcher, R.: A nonmonotone filter method for nonlinear optimization. Comput. Optim. Appl. 52, 583–607 (2012)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Su, K., Pu, D.: A nonmonotone filter trust region method for nonlinear constrained optimization. J. Comput. Appl. Math. 223, 230–239 (2009)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Su, K., Yu, Z.: A modified SQP method with nonmonotone technique and its global convergence. Comput. Math. Appl. 57, 240–247 (2009)

    MathSciNet  MATH  Google Scholar 

  45. 45.

    Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods, and Applications. Graduate Studies in Mathematics. American Mathematical Society, Providence (2010)

    MATH  Google Scholar 

  46. 46.

    Ulbrich, M., Ulbrich, S.: Non-monotone trust region methods for nonlinear equality constrained optimization without a penalty function. Math. Program. 95(2003), 103–135 (2003)

    MathSciNet  MATH  Google Scholar 

  47. 47.

    Vallejos, M.: Multigrid methods for elliptic optimal control problems with pointwise state constraints. Numer. Math. Theory Methods Appl. 5, 99–109 (2012)

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Vallejos, M.: A comparison of smoothers for state-constrained optimal control problems. Philipp. Sci. Lett. 7, 13–21 (2014)

    Google Scholar 

  49. 49.

    Zhang, H.C., Hager, W.W.: A nonmonotone line search technique and its application to unconstrained optimization. SIAM J. Optim. 14, 1043–1056 (2004)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Omid Solaymani Fard.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Davod Khojasteh Salkuyeh.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fard, O.S., Sarani, F. & Nosratipour, H. An Efficient Nonmonotone Method for State-Constrained Elliptic Optimal Control Problems. Bull. Iran. Math. Soc. 46, 943–963 (2020). https://doi.org/10.1007/s41980-019-00303-6

Download citation

Keywords

  • Optimal control
  • Nonmonotone semismooth Newton method
  • State constraints
  • Finite difference discretization scheme

Mathematics Subject Classification

  • 49K20
  • 65N06
  • 90C30