Acta Mathematica Vietnamica

, Volume 43, Issue 1, pp 175–199

# The Mordukhovich Coderivative and the Local Metric Regularity of the Solution Map to a Parametric Discrete Optimal Control Problem

• Le Quang Thuy
• Nguyen Thi Toan
Article

## Abstract

In this paper, we study the Mordukhovich coderivative and the local metric regularity in Robinson’s sense of the solution map to a parametric dynamic programming problem with linear constraints and convex cost functions. By establishing abstract results on the coderivative and the local metric regularity of the solution map to a parametric variational inequality, we obtain the Mordukhovich coderivative and the local metric regularity in Robinson’s sense of the solution map to a parametric discrete optimal control problem.

## Keywords

Parametric discrete optimal control problem Dynamic programming problem Solution map Local metric regularity Mordukhovich coderivative

## Mathematics Subject Classification (2010)

49J21 49K21 93C55

## Notes

### Acknowledgements

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2015.04 and by the Vietnam Institute for Advanced Study in Mathematics (VIASM).

## References

1. 1.
Arutyunov, A.V., Marinkovich, B.: Necessary optimality conditions for discrete optimal control problems. Mosc. Univ. Comput. Math. Cybern. 1, 38–44 (2005)
2. 2.
Bertsekas, D.P.: Dynamic Programming and Optimal Control, vol. I. Springer, Berlin (2005)
3. 3.
Gabasov, R., Mordukhovich, B.S., Kirillova, F.M.: The discrete maximum principle. Doklady Akademii Nauk SSSR 213, 19–22 (1973). (Russian; English transl. in Soviet Math. Dokl. 14, 1624–1627 (1973))
4. 4.
Henrion, R., Mordukhovich, B.S., Nam, N.M.: Second-order analysis of polyhedral systems in finite dimensions with applications to robust stability of variational inequalities. SIAM J. Optim. 20, 2199–2227 (2010)
5. 5.
Larson, R.E., Casti, J.: Principles of Dynamic Programming, vol. I. Marcel Dekker, New York (1982)
6. 6.
Larson, R.E., Casti, J.: Principles of Dynamic Programming, vol. II. Marcel Dekker, New York (1982)
7. 7.
Lian, Z., Liu, L., Neuts, M.F.: A discrete-time model for common lifetime inventory systems. Math. Oper. Res. 30, 718–732 (2005)
8. 8.
Malanowski, K.: Differential sentivity of solutions of convex constrained optimal control problems for discrete systems. J. Optim. Theory Appl. 53, 429–449 (1987)
9. 9.
Mangasarian, O.L., Shiau, T.-H.: Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems. SIAM J. Control Optim. 25, 583–595 (1987)
10. 10.
Mordukhovich, B.S.: Difference approximations of optimal control system. Prikladaya Matematika I Mekhanika 42, 431–440 (1978). (Russian; English transl. in J. Appl. Math. Mech. 42, 452–461 (1978))
11. 11.
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I. Basis Theory. Springer, Berlin (2006)Google Scholar
12. 12.
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation II. Applications. Springer, Berlin (2006)Google Scholar
13. 13.
Mordukhovich, B.S.: Generalized differential calculus for nonsmooth and set-value mappings. J. Math. Anal. Appl. 183, 250–288 (1994)
14. 14.
Nam, N.M.: Coderivatives of normal cone mappings and Lipschitzian stability of parametric variational inequalities. Non. Anal. 73, 2271–2281 (2010)
15. 15.
Pindyck, R.S.: An aplication of the linear quaratic tracking problem to economic stabilization policy. IEEE Tran. Automatic Con. 17, 287–300 (1972)
16. 16.
Quy, N.T.: New results on linearly perturbed polyhedral normal cone mapping. J. Math. Anal. Appl. 381, 352–364 (2011)
17. 17.
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)
18. 18.
Seeger, A.: Subgradient of optimal-value function in dynamic programming: The case of convex system without optimal paths. Math. Oper. Res. 21, 555–575 (1996)
19. 19.
Toan, N.T., Kien, B.T.: Continuity properties of the solution map to a parametric discrete optimal control problem. J. Non. Conv. Anal. 12, 635–650 (2011)
20. 20.
Toan, N.T., Yao, J.-C.: Mordukhovich subgradients of the value function to a parametric discrete optimal control problem. J. Glob. Optim. 58, 595–612 (2014)
21. 21.
Toan, N.T., Ansari, Q.H., Yao, J.-C.: Second-order necessary optimality conditions for a discrete optimal control problem. J. Optim. Theory Appl. 165(3), 812–836 (2015)
22. 22.
Toan, N.T., Thuy, L.Q.: Second-order necessary optimality conditions for a discrete optimal control problem with mixed constraints. J. Glob. Optim. 64, 533–562 (2016)
23. 23.
Tu, P.N.V.: Introductory Optimization Dynamics. Springer, Berlin (1991)
24. 24.
Yen, N.D., Yao, J.-C.: Point-based sufficient conditions for metric regularity of implicit multifunctions. Non. Anal. 70, 2806–2815 (2009)