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On sufficient optimality conditions for multiobjective control problems

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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.

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Notes

  1. A problem is said to be invex if the defining functions are invex with the same invexity kernel.

References

  1. Antczak, T.: On G-invex multiobjective programming. Part I. Optimality. J. Glob. Optim. 43, 97–109 (2009)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arana-Jiménez, M., Osuna-Gómez, R., Rufián-Lizana, A., Ruiz-Garzón, G.: KT-invex control problem. Appl. Math. Comput. 197, 489–496 (2008)

    MathSciNet  MATH  Google Scholar 

  5. Arana-Jiménez, M., Hernández-Jiménez, B., Ruiz-Garzón, G., Rufián-Lizana, A.: FJ-invex control problem. Appl. Math. Lett. 22, 1887–1891 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bellaassali, S., Jourani, A.: Necessary optimality conditions in multiobjective dynamic optimization. SIAM J. Control Optim. 42, 2043–2061 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  14. Burai, P.: Necessary and sufficient condition on global optimality without convexity and second order differentiability. Optim. Lett. 7(5), 903–911 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Burai, P.: Local-global minimum property in unconstrained minimization problems. J. Optim. Theory Appl. 162(1), 34–46 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Clarke, F.H.: Optimization and Nonsmooth Analysis, Classics in Applied Mathematics, vol. 5. SIAM, Philadelphia (1990)

    Book  Google Scholar 

  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. Clarke, F.: Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics, vol. 264. Springer, London (2013)

    Book  Google Scholar 

  19. Craven, B.D., Glover, B.M.: Invex functions and duality. J. Aust. Math. Soc. Ser. A 39, 1–20 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. de Oliveira, V.A., Rojas-Medar, M.A.: Continuous-time optimization problems involving invex functions. J. Math. Anal. Appl. 327, 1320–1334 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. de Oliveira, V.A., Rojas-Medar, M.A.: Multi-objective infinite programming. Comput. Math. Appl. 55, 1907–1922 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. de Oliveira, V.A., Silva, G.N.: New optimality conditions for nonsmooth control problems. J. Glob. Optim. 57, 1465–1484 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  30. Hanson, M.A.: On sufficiency of the Kuhn–Tucker conditions. J. Math. Anal. Appl. 80, 545–550 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hanson, M.A.: Invexity and the Kuhn–Tucker theorem. J. Math. Anal. Appl. 236, 594–604 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

  38. Kenan, Z., Lok, T.: Power control for uplink transmission with mobile users. IEEE Trans. Veh. Technol. 60(5), 2117–2127 (2011)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  41. Martin, D.H.: The essence of invexity. J. Optim. Theory Appl. 47, 65–76 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  43. Mishra, S.K., Wang, S., Lai, K.K.: Generalized Convexity and Vector Optimization, Nonconvex Optimization and Its Applications, vol. 90. Springer, Berlin (2009)

    MATH  Google Scholar 

  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)

    Book  Google Scholar 

  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. Nickisch, H., Seeger, M.: Multiple kernel learning: A unifying probabilistic viewpoint (2011). arXiv:1103.0897

  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)

    Article  MathSciNet  MATH  Google Scholar 

  48. Osuna-Gómez, R., Beato-Moreno, A., Rufián-Lizana, A.: Generalized convexity in multiobjective programming. J. Math. Anal. Appl. 233, 205–220 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  51. Reiland, T.W.: Nonsmooth invexity. Bull. Aust. Math. Soc. 42, 437–446 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  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. Sach, P.H., Kim, D.S., Lee, G.M.: Generalized convexity and nonsmooth problems of vector optimization. J. Glob. Optim. 31, 383–403 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  54. Silva, G.N., Vinter, R.B.: Necessary conditions for optimal impulsive control problems. SIAM J. Control Optim. 35(6), 1829–1846 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  55. Slimani, H., Radjef, M.S.: Multiobjective Programming Under Generalized Invexity. Lap Lambert Academic Publishing, Saarbrücken (2010)

  56. Slimani, H., Radjef, M.S.: Nondifferentiable multiobjective programming under generalized \(d_I\)-invexity. Eur. J. Oper. Res. 202(1), 32–41 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  57. Syed, M., Pardalos, P., Principe, J.: Invexity of the minimum error entropy criterion. IEEE Signal Process. Lett. 20(12), 1159–1162 (2013)

    Article  Google Scholar 

  58. Vinter, R.B.: Optim. Control. Birkhäuser, Boston (2000)

    Google Scholar 

  59. Zhao, F.A.: On sufficiency of the Kuhn–Tucker conditions in nondifferentiable programming. Bull. Aust. Math. Soc. 46, 385–389 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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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).

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  1. Instituto de Biociências, Letras e Ciências Exatas, Departamento de Matemática Aplicada, UNESP - Universidade Estadual Paulista, Câmpus de São José do Rio Preto, Rua Cristóvão Colombo, N. 2265, São José do Rio Preto, SP, CEP 15054-000, Brazil

    Valeriano Antunes de Oliveira & Geraldo Nunes Silva

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  1. Valeriano Antunes de Oliveira
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  2. Geraldo Nunes Silva
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Correspondence to Valeriano Antunes de Oliveira.

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de Oliveira, V.A., Silva, G.N. On sufficient optimality conditions for multiobjective control problems. J Glob Optim 64, 721–744 (2016). https://doi.org/10.1007/s10898-015-0351-y

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