Advertisement

Journal of Global Optimization

, Volume 64, Issue 4, pp 721–744 | Cite as

On sufficient optimality conditions for multiobjective control problems

  • Valeriano Antunes de OliveiraEmail author
  • Geraldo Nunes Silva
Article

Abstract

This paper is devoted to presenting optimality conditions for the sufficiency of the maximum principle for multiobjective optimal control problems with nonsmooth data. Such conditions are the most general as possible in the sense that problems in which the set of necessary conditions from the maximum principle are also sufficient, necessarily obey them. A variation of such conditions is also presented, under which the set of optimal solutions of the multiobjective problem can be determined by resolving a related scalar weighting problem.

Keywords

Optimal control Optimality conditions Generalized convexity Multiobjective programming 

Mathematics Subject Classification

49K15 90C26 90C29 

Notes

Acknowledgments

The authors were supported by Grant 2013/07375-0, São Paulo Research Foundation (FAPESP) and by Grants 457785/2014-4, 479109/2013-3, and 309335/2012-4, National Council for Scientific and Technological Development (CNPq).

References

  1. 1.
    Antczak, T.: On G-invex multiobjective programming. Part I. Optimality. J. Glob. Optim. 43, 97–109 (2009)Google Scholar
  2. 2.
    Antczak, T.: Proper efficiency conditions and duality results for nonsmooth vector optimization in Banach spaces under \((\varPhi ,\rho )\)-invexity. Nonlin. Anal. 75(6), 3107–3121 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arana-Jiménez, M., Osuna-Gómez, R., Ruiz-Garzón, G., Rojas-Medar, M.: On variational problems: characterization of solutions and duality. J. Math. Anal. Appl. 311, 1–12 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Arana-Jiménez, M., Osuna-Gómez, R., Rufián-Lizana, A., Ruiz-Garzón, G.: KT-invex control problem. Appl. Math. Comput. 197, 489–496 (2008)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Arana-Jiménez, M., Hernández-Jiménez, B., Ruiz-Garzón, G., Rufián-Lizana, A.: FJ-invex control problem. Appl. Math. Lett. 22, 1887–1891 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Arana-Jiménez, M., Rufián-Lizana, A., Ruiz-Garzón, G., Osuna-Gómez, R.: Efficient solutions in V-KT-pseudoinvex multiobjective control problems: a characterization. Appl. Math. Comput. 215, 441–448 (2009)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Arana-Jiménez, M., Ruiz-Garzón, G., Osuna-Gómez, R., Rufián-Lizana, A.: Weakly efficient solutions and pseudoinvexity in multi objective control problems. Nonlin. Anal. 73, 1792–1801 (2010)CrossRefzbMATHGoogle Scholar
  8. 8.
    Arana-Jiménez, M., Ruiz-Garzón, G., Rufián-Lizana, A., Osuna-Gómez, R.: Weak efficiency in multiobjective variational problems under generalized convexity. J. Glob. Optim. 52, 109–121 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Arana-Jiménez, M., Ruiz-Garzón, G., Osuna-Gómez, R., Hernández-Jiménez, B.: Duality and a characterization of pseudoinvexity for Pareto and weak Pareto solutions in nondifferentiable multiobjective programming. J. Optim. Theory Appl. 156, 266–277 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Arutyunov, A.V., Karamzin, D.Y., Pereĭra, F.: R. V. Gamkrelidze’s maximum principle for optimal control problems with bounded phase coordinates and its relation to other optimality conditions. Dokl. Akad. Nauk 436(6), 738–742 (2011)zbMATHGoogle Scholar
  11. 11.
    Arutyunov, A.V., Karamzin, D.Y., Pereira, F.L.: The maximum principle for optimal control problems with state constraints by R. V. Gamkrelidze: revisited. J. Optim. Theory Appl. 149(3), 474–493 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bellaassali, S., Jourani, A.: Necessary optimality conditions in multiobjective dynamic optimization. SIAM J. Control Optim. 42, 2043–2061 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bonnel, H., Kaya, C.Y.: Optimization over the efficient set of multi-objective convex optimal control problems. J. Optim. Theory Appl. 147(1), 93–112 (2010)Google Scholar
  14. 14.
    Burai, P.: Necessary and sufficient condition on global optimality without convexity and second order differentiability. Optim. Lett. 7(5), 903–911 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Burai, P.: Local-global minimum property in unconstrained minimization problems. J. Optim. Theory Appl. 162(1), 34–46 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Clarke, F.H.: Optimization and Nonsmooth Analysis, Classics in Applied Mathematics, vol. 5. SIAM, Philadelphia (1990)CrossRefGoogle Scholar
  17. 17.
    Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, vol. 178. Springer, New York (1998)Google Scholar
  18. 18.
    Clarke, F.: Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics, vol. 264. Springer, London (2013)CrossRefGoogle Scholar
  19. 19.
    Craven, B.D., Glover, B.M.: Invex functions and duality. J. Aust. Math. Soc. Ser. A 39, 1–20 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    de Oliveira, V.A., Rojas-Medar, M.A.: Continuous-time multiobjective optimization problems via invexity. Abstr. Appl. Anal. 2007, Art. ID 61296, 11 (2007)Google Scholar
  21. 21.
    de Oliveira, V.A., Rojas-Medar, M.A., Brandão, A.J.V.: A note on KKT-invexity in nonsmooth continuous-time optimization. Proyecciones 26, 269–279 (2007)MathSciNetzbMATHGoogle Scholar
  22. 22.
    de Oliveira, V.A., Silva, G.N., Rojas-Medar, M.A.: A class of multiobjective control problems. Optim. Control Appl. Methods 30, 77–86 (2009)MathSciNetCrossRefGoogle Scholar
  23. 23.
    de Oliveira, V.A., Silva, G.N., Rojas-Medar, M.A.: KT-invexity in optimal control problems. Nonlin. Anal. Theory Methods Appl. 71, 4790–4797 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    de Oliveira, V.A., dos Santos, L.B., Osuna-Gómez, R., Rojas-Medar, M.A.: Optimality conditions for nonlinear infinite programming problems. Optim. Lett. 9, 1131–1147 (2015). doi: 10.1007/s11590-014-0808-9
  25. 25.
    de Oliveira, V.A., Rojas-Medar, M.A.: Continuous-time optimization problems involving invex functions. J. Math. Anal. Appl. 327, 1320–1334 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    de Oliveira, V.A., Rojas-Medar, M.A.: Multi-objective infinite programming. Comput. Math. Appl. 55, 1907–1922 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    de Oliveira, V.A., Silva, G.N.: New optimality conditions for nonsmooth control problems. J. Glob. Optim. 57, 1465–1484 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Dinuzzo, F., Ong, C., Gehler, P., Pillonetto, G.: Learning output kernels with block coordinate descent. In: Proceedings of the 28th International Conference on Machine Learning, ICML 2011, pp. 49–56 (2011)Google Scholar
  29. 29.
    Gamkrelidze, R.V.: Principles of optimal control theory (transl. from the Russian by K. Malowski. Transl. ed. by and with a foreword by L. D. Berkovitz). In Mathematical Concepts and Methods in Science and Engineering, vol. 7. Plenum Press, New York, London (1978)Google Scholar
  30. 30.
    Hanson, M.A.: On sufficiency of the Kuhn–Tucker conditions. J. Math. Anal. Appl. 80, 545–550 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Hanson, M.A.: Invexity and the Kuhn–Tucker theorem. J. Math. Anal. Appl. 236, 594–604 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Hernández-Jiménez, B., Rojas-Medar, M.A., Osuna-Gómez, R., Beato-Moreno, A.: Generalized convexity in non-regular programming problems with inequality-type constraints. J. Math. Anal. Appl. 352, 604–613 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Hernández-Jiménez, B., Osuna-Gómez, R., Arana-Jiménez, M., Ruiz-Garzón, G.: Generalized convexity and efficiency for non-regular multiobjective programming problems with inequality-type constraints. Nonlin. Anal. Theory Methods Appl. 73, 2463–2475 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Hernández-Jiménez, B., Rojas-Medar, M.A., Osuna-Gómez, R., Rufián-Lizana, A.: Characterization of weakly efficient solutions for non-regular multiobjective programming problems with inequality-type constraints. J. Convex Anal. 18, 749–768 (2011)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Hernández-Jiménez, B., Osuna-Gómez, R., Rojas-Medar, M.A., dos Santos, L.B.: Generalized convexity for non-regular optimization problems with conic constraints. J. Glob. Optim. 57, 649–662 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Karamzin, D.Y., de Oliveira, V.A., Pereira, F.L., Silva, G.N.: On some extension of optimal control theory. Eur. J. Control 20(6), 284–291 (2014)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Kenan, Z., Lok, T.: Optimal power allocation for relayed transmission through a mobile relay node. In: IEEE Vehicular Technology Conference. Article number 5494018 (2010)Google Scholar
  38. 38.
    Kenan, Z., Lok, T.: Power control for uplink transmission with mobile users. IEEE Trans. Veh. Technol. 60(5), 2117–2127 (2011)CrossRefGoogle Scholar
  39. 39.
    Kien, B.T., Wong, N.C., Yao, J.C.: Necessary conditions for multiobjective optimal control problems with free end-time. SIAM J. Control Optim. 47(5), 2251–2274 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Luc, D.T.: Generalized convexity in vector optimization. In: Hadjisavvas, N., Komlsi, S., Schaible, S. (eds.) Handbook of Generalized Convexity and Generalized Monotonicity, Nonconvex Optimization and Its Applications, vol. 76, pp. 195–236. Springer, New York (2005)CrossRefGoogle Scholar
  41. 41.
    Martin, D.H.: The essence of invexity. J. Optim. Theory Appl. 47, 65–76 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Mishra, S.K., Wang, S., Lai, K.K.: V-Invex Functions and Vector Optimization, Springer Optimization and Its Applications, vol. 14. Springer, New York (2008)zbMATHGoogle Scholar
  43. 43.
    Mishra, S.K., Wang, S., Lai, K.K.: Generalized Convexity and Vector Optimization, Nonconvex Optimization and Its Applications, vol. 90. Springer, Berlin (2009)zbMATHGoogle Scholar
  44. 44.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I. Basic Theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 330. Springer, Berlin (2006)CrossRefGoogle Scholar
  45. 45.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation II. Applications, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 331. Springer, Berlin (2006)Google Scholar
  46. 46.
    Nickisch, H., Seeger, M.: Multiple kernel learning: A unifying probabilistic viewpoint (2011). arXiv:1103.0897
  47. 47.
    Osuna-Gómez, R., Rufián-Lizana, A., Ruíz-Canales, P.: Invex functions and generalized convexity in multiobjective programming. J. Optim. Theory Appl. 98, 651–661 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Osuna-Gómez, R., Beato-Moreno, A., Rufián-Lizana, A.: Generalized convexity in multiobjective programming. J. Math. Anal. Appl. 233, 205–220 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The mathematical theory of optimal processes (translated by D. E. Brown. A Pergamon Press Book). The Macmillan Co., New York (1964)Google Scholar
  50. 50.
    Pontryagin, L., Boltyanskij, V., Gamkrelidze, R., Mishchenko, E.: Selected works, vol. 4. The mathematical theory of optimal processes. Ed. and with a preface by R. V. Gamkrelidze (transl. from the Russian by K. N. Trirogoff. Transl. ed. by L. W. Neustadt. With a preface by L. W. Neustadt and K. N. Trirogoff. Reprint of the 1962 Engl. translation. Classics of Soviet Mathematics). Gordon and Breach Science Publishers, New York, p. xxiv (1986)Google Scholar
  51. 51.
    Reiland, T.W.: Nonsmooth invexity. Bull. Aust. Math. Soc. 42, 437–446 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317. Springer, Berlin (1998)Google Scholar
  53. 53.
    Sach, P.H., Kim, D.S., Lee, G.M.: Generalized convexity and nonsmooth problems of vector optimization. J. Glob. Optim. 31, 383–403 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Silva, G.N., Vinter, R.B.: Necessary conditions for optimal impulsive control problems. SIAM J. Control Optim. 35(6), 1829–1846 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Slimani, H., Radjef, M.S.: Multiobjective Programming Under Generalized Invexity. Lap Lambert Academic Publishing, Saarbrücken (2010)Google Scholar
  56. 56.
    Slimani, H., Radjef, M.S.: Nondifferentiable multiobjective programming under generalized \(d_I\)-invexity. Eur. J. Oper. Res. 202(1), 32–41 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Syed, M., Pardalos, P., Principe, J.: Invexity of the minimum error entropy criterion. IEEE Signal Process. Lett. 20(12), 1159–1162 (2013)CrossRefGoogle Scholar
  58. 58.
    Vinter, R.B.: Optim. Control. Birkhäuser, Boston (2000)Google Scholar
  59. 59.
    Zhao, F.A.: On sufficiency of the Kuhn–Tucker conditions in nondifferentiable programming. Bull. Aust. Math. Soc. 46, 385–389 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Zhu, Q.J.: Hamiltonian necessary conditions for a multiobjective optimal control problem with endpoint constraints. SIAM J. Control Optim. 39(1), 97–112 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Valeriano Antunes de Oliveira
    • 1
    Email author
  • Geraldo Nunes Silva
    • 1
  1. 1.Instituto de Biociências, Letras e Ciências Exatas, Departamento de Matemática AplicadaUNESP - Universidade Estadual PaulistaSão José do Rio PretoBrazil

Personalised recommendations