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

  • Omid Solaymani FardEmail author
  • Farhad Sarani
  • Hadi Nosratipour
Original Paper


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.


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

Mathematics Subject Classification

49K20 65N06 90C30 



  1. 1.
    Ahookhosh, M., Ghaderi, S.: On efficiency of nonmonotone Armijo-type line searches. Appl. Math. Model. 43, 170–190 (2017)MathSciNetCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)zbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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). CrossRefGoogle Scholar
  7. 7.
    Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms. Wiley, New York (2006)zbMATHCrossRefGoogle Scholar
  8. 8.
    Bergounioux, M., Kunisch, K.: Primal-dual strategy for state-constrained optimal control problems. Comput. Optim. Appl. 22, 193–224 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bonettini, S.: A nonmonotone inexact Newton method. Optim. Methods Softw. 20, 475–491 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Borzì, A.: Smoothers for control- and state-constrained optimal control problems. Comput. Vis. Sci. 11, 59–66 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Borzì, A., Schulz, V.: Computational Optimization of Systems Governed by Partial Differential Equations. Computational Science and Engineering. SIAM, Philadelphia (2012)zbMATHGoogle Scholar
  12. 12.
    Cantrell, S., Cosner, C., Ruan, S.: Spatial Ecology, CRC Mathematical and Computational Biology. CRC Press, Boca Raton (2009)CrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Christofides, P., Armaou, A., Lou, Y., Varshney, A.: Control and Optimization of Multiscale Process Systems, Control Engineering. Birkhäuser, Boston (2008)zbMATHGoogle Scholar
  18. 18.
    Diehl, M., Glineur, F., Jarlebring, E., Michiels, W.: Recent Advances in Optimization and its Applications in Engineering. Springer, Berlin (2010)CrossRefGoogle 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)CrossRefGoogle Scholar
  20. 20.
    Field, D.A., Komkov, V.: Theoretical aspects of industrial design. In: Proceedings in Applied Mathematics Series. SIAM (1992)Google Scholar
  21. 21.
    Griva, I., Nash, S.G., Sofer, A.: Linear and Nonlinear Optimization, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2009)zbMATHCrossRefGoogle Scholar
  22. 22.
    Hinze, M., Pinnau, R., Ulbrich, R., Ulbrich, S.: Optimization with PDE Constraints. Springer, Berlin (2008)zbMATHGoogle 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)Google Scholar
  24. 24.
    Ito, K., Kunisch, K.: Lagrange Multiplier Approach to Variational Problems and Applications. Society for Industrial and Applied Mathematics, Philadelphia (2008)zbMATHCrossRefGoogle Scholar
  25. 25.
    Kimiaei, M., Rahpeymaii, F.: A new nonmonotone linesearch trust-region approach for nonlinear systems. TOP 27, 199–232 (2019)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, New York (1971)zbMATHCrossRefGoogle Scholar
  29. 29.
    Luenberger, D.G., Ye, Y.: Linear and Nonlinear Programming. Springer, New York (2008)zbMATHCrossRefGoogle 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)zbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)zbMATHCrossRefGoogle 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). CrossRefzbMATHGoogle Scholar
  36. 36.
    Nosratipour, H., Borzabadi, A.H., Fard, O.S.: On the nonmonotonicity degree of nonmonotone line searches. Calcolo 54, 1217–1242 (2017b)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Pang, J.S., Qi, L.: Nonsmooth equations: motivation and algorithms. SIAM J. Optim. 3, 443–465 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Shen, C., Leyffer, S., Fletcher, R.: A nonmonotone filter method for nonlinear optimization. Comput. Optim. Appl. 52, 583–607 (2012)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods, and Applications. Graduate Studies in Mathematics. American Mathematical Society, Providence (2010)zbMATHGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Mathematical SciencesFerdowsi University of MashhadMashhadIran
  2. 2.School of Mathematics and Computer ScienceDamghan UniversityDamghanIran
  3. 3.Department of Mathematics, Faculty of ScienceRazi UniversityKermanshahIran

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