Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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OPTIMALITY CONDITIONS AND DUALITY MODELS FOR MINMAX FRACTIONAL OPTIMAL CONTROL PROBLEMS CONTAINING ARBITRARY NORMS

  • G. J., Zalmai (Department of Mathematics and Computer Science Northern Michigan University)
  • Published : 2004.09.01
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Abstract

Both parametric and parameter-free necessary and sufficient optimality conditions are established for a class of nondiffer-entiable nonconvex optimal control problems with generalized fractional objective functions, linear dynamics, and nonlinear inequality constraints on both the state and control variables. Based on these optimality results, ten Wolfe-type parametric and parameter-free duality models are formulated and weak, strong, and strict converse duality theorems are proved. These duality results contain, as special cases, similar results for minmax fractional optimal control problems involving square roots of positive semi definite quadratic forms, and for optimal control problems with fractional, discrete max, and conventional objective functions, which are particular cases of the main problem considered in this paper. The duality models presented here contain various extensions of a number of existing duality formulations for convex control problems, and subsume continuous-time generalizations of a great variety of similar dual problems investigated previously in the area of finite-dimensional nonlinear programming.

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References

  1. D. J. Ashton, and D. R. Atkins, Multicriteria programming for financial planning, J. Oper. Res. Soc. Japan 30 (1979), 259–270
  2. I. Barrodale, Best rational approximation and strict quasiconvexity, SIAM J. Numer. Anal. 10 (1973), 8–12 https://doi.org/10.1137/0710002
  3. I. Barrodale, M. J. D. Powell and F. D. K. Roberts, The differential correction algorithm for rational ${\ell}_{\infty}$-approximation, SIAM J. Numer. Anal. 9 (1972), 493–504. https://doi.org/10.1137/0709044
  4. S. Chandra, B. D. Craven and I. Husain, A class of nondifferentiable control problems, J. Optim. Theory Appl. 56 (1985), 227–243 https://doi.org/10.1007/BF00939409
  5. S. Chandra, B. D. Craven and I. Husain, Continuous programming containing arbitrary norms, J. Aust. Math. Soc. 39 (1985), 28–38 https://doi.org/10.1017/S144678870002214X
  6. S. Chandra, B. D. Craven and I. Husain, A class of nondifferentiable continuous programming problems, J. Math. Anal. Appl. 107 (1985), 122–131. https://doi.org/10.1016/0022-247X(85)90357-9
  7. A. Charnes and W. W. Cooper, Goal programming and multiobjective optimization, part 1, European J. Oper. Res. 1 (1977), 39–54. https://doi.org/10.1016/S0377-2217(77)81007-2
  8. B. D. Craven, (1988) Fractional Programming, Heldermann Verlag, Berlin, 1988
  9. V. F. Demyanov and V. N. Malozemov, Introduction to Minimax, Wiley, New York, 1974
  10. W. Dinkelbach, On nonlinear fractional programming, Management Sci. 13 (1967), 492–498 https://doi.org/10.1287/mnsc.13.7.492
  11. I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1976
  12. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1985
  13. J. S. H. Kornbluth, A survey of goal programming, Omega 1 (1973), 193–205 https://doi.org/10.1016/0305-0483(73)90023-6
  14. W. Krabs, Dualität bei diskreter rationaler Approximation, Internat. Ser. Numer. Math. 7 (1967), 33–41
  15. J. V. Outrata and J. Jarusek, Duality in mathematical programming and optimal control, Kybernetika 20 (1984), 1–119
  16. R. T. Rockafellar, Conjugate functions in optimal control and the calculus of variations, J. Math. Anal. Appl. 32 (1970), 174–222 https://doi.org/10.1016/0022-247X(70)90324-0
  17. R. T. Rockafellar, Duality in optimal control, Lecture Notes in Mathematics, Vol. 680, Springer-Verlag, Berlin, 1978, pp. 219–257
  18. I. M. Stancu-Minasian, Fractional Programming : Theory, Methods, and Applications, Kluwer Academic Publishers, Boston, 1977
  19. M. B. Subrahmanyam, On applications of control theory to integral inequalities, J. Math. Anal. Appl. 77 (1980), 47–59 https://doi.org/10.1016/0022-247X(80)90260-7
  20. M. B. Subrahmanyam, Optimal Control with a Worst-Case Performance Criterion and Applications, Lecture Notes in Control and Inform. Sci. 145, Springer-Verlag, Berlin, 1990
  21. M. B. Subrahmanyam, Synthesis of finite-interval H_{\infty} controllers by state-space methods, J. Guidance Control Dynamics 13 (1990), 624–627 https://doi.org/10.2514/3.25379
  22. M. B. Subrahmanyam, Optimal disturbance rejection and performance robustness in linear systems, J. Math. Anal. Appl. 164 (1992), 130–150 https://doi.org/10.1016/0022-247X(92)90149-8
  23. J. Von Neumann, A model of general economic equilibrium, Rev. Econom. Stud. 13 (1945), 1–9 https://doi.org/10.2307/2296111
  24. J. Werner, Duality in generalized fractional programming, Internat. Ser. Numer. Math. 84 (1988), 341–351
  25. G. J. Zalmai, A transposition theorem with applications to optimal control problems, Optimization 29 (1989), 265–279 https://doi.org/10.1080/02331938908843441
  26. G. J. Zalmai, Optimality conditions and duality models for a class of nonsmooth constrained fractional variational problems, Optimization 30 (1994), 15–51 https://doi.org/10.1080/02331939408843969
  27. G. J. Zalmai, Proper efficiency conditions and duality models for a class of constrained multiobjective fractional optimal control problems containing arbitrary norms, J. Optim. Theory Appl. 90 (1996), 435–456 https://doi.org/10.1007/BF02190007
  28. G. J. Zalmai, Optimality conditions and duality models for a class of nonsmooth constrained fractional optimal control problems, J. Math. Anal. Appl. 210 (1997), 114–149 https://doi.org/10.1006/jmaa.1997.5377
  29. G. J. Zalmai, Saddle-point-type optimality conditions and Lagrangian-type duality for a class of constrained generalized fractional optimal control problems, Optimization 44 (1998), 351–372 https://doi.org/10.1080/02331939808844417
  30. G. J. Zalmai, Optimality principles and duality models for a class of continuous-time generalized fractional programming problems with operator constraints, J. Stat. Manag. Syst. 1 (1998), 61–100

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