Differential stability of convex discrete optimal control problems in Banach spaces is studied in this paper. By using some recent results of An and Yen (Appl. Anal. 94, 108–128, 2015) on differential stability of parametric convex optimization problems under inclusion constraints, we obtain an upper estimate for the subdifferential of the optimal value function of a parametric convex discrete optimal control problem, where the objective function may be nondifferentiable. If the objective function is differentiable, the obtained upper estimate becomes an equality. It is shown that the singular subdifferential of the just mentioned optimal value function always consists of the origin of the dual space.
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An, D. T. V., Yao, J.-C.: Further results on differential stability of convex optimization problems. J. Optim. Theory Appl. 170, 28–42 (2016)
An, D. T. V., Yen, N. D.: Differential stability of convex optimization problems under inclusion constraints. Appl. Anal. 94, 108–128 (2015)
Bartl, D.: A short algebraic proof of the Farkas lemma. SIAM J. Optim. 19, 234–239 (2008)
Bertsekas, D. P.: Dynamic Programming and Optimal Control, vol. I. Athena Scientific, Belmont (2005)
Bonnans, J. F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)
Chieu, N. H., Kien, B. T., Toan, N. T.: Further results on subgradients of the value function to a parametric optimal control problem. J. Optim. Theory Appl. 168, 785–801 (2016)
Chieu, N. H., Yao, J. -C.: Subgradients of the optimal value function in a parametric discrete optimal control problem. J. Ind. Manag. Optim. 6, 401–410 (2010)
Ioffe, A. D., Tihomirov, V. M.: Theory of Extremal Problems. North-Holland Publishing Company, North-Holland (1979)
Kien, B. T., Liou, Y. C., Wong, N. -C., Yao, J. -C.: Subgradients of value functions in parametric dynamic programming. European J. Oper. Res. 193, 12–22 (2009)
Mordukhovich, B. S.: Variational Analysis and Generalized Differentiation. Volume I: Basic Theory, Volume II: Applications. Springer, Berlin (2006)
Mordukhovich, B. S., Nam, N. M., Yen, N. D.: Subgradients of marginal functions in parametric mathematical programming. Math. Program. Ser. B 116, 369–396 (2009)
Moussaoui, M., Seeger, A.: Sensitivity analysis of optimal value functions of convex parametric programs with possibly empty solution sets. SIAM J. Optim. 4, 659–675 (1994)
Rockafellar, R. T.: Hamilton-Jacobi theory and parametric analysis in fully convex problems of optimal control. J. Global Optim. 28, 419–431 (2004)
Seeger, A.: Subgradients of optimal-value functions in dynamic programming: the case of convex systems without optimal paths. Math. Oper. Res. 21, 555–575 (1996)
Thuy, L. Q., Toan, N. T.: Subgradients of the value function in a parametric convex optimal control problem. J. Optim. Theory Appl. 170, 43–64 (2016)
Toan, N.: Mordukhovich subgradients of the value function in a parametric optimal control problem. Taiwanese J. Math. 19, 1051–1072 (2015)
Toan, N. T., Kien, B. T.: Subgradients of the value function to a parametric optimal control problem. Set-Valued Var. Anal. 18, 183–203 (2010)
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)
Tu, P. N. V.: Introductory Optimization Dynamics. Springer-Verlag, Berlin (1984)
Vinter, R.: Optimal Control. Birkhäuser, Boston (2000)
The research of Duong Thi Viet An was supported by the College of Sciences, Thai Nguyen University, Vietnam. The research of Nguyen Thi Toan was supported by the National Foundation for Science and Technology Development (Vietnam) under grant number 101.01-2015.04. The authors thank Professor Nguyen Dong Yen for useful discussions and the anonymous referees for valuable remarks.
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An, D.T.V., Toan, N. Differential Stability of Convex Discrete Optimal Control Problems. Acta Math Vietnam 43, 201–217 (2018). https://doi.org/10.1007/s40306-017-0227-y
- Parametric convex discrete optimal control problem
- Optimal value function
- Linear operator with closed range
- Adjoint operator
Mathematics Subject Classification (2010)