Motivated by the works of Toan and Kien (J. Nonlinear Convex Anal.12:635–650, 2011) and Thuy and Toan (Acta Math. Vietnam.43:175–199, 2018) on the stability and the solution sensitivity in parametric discrete optimal control problems, this paper is devoted to the study of the Lipschitz property of the value function and the Aubin property of the solution map to a parametric dynamic programming problem with linear constraints. By establishing abstract results on the Lipschitz property of the value function to a parametric programming and the Aubin property of the solution map to a parametric variational inequality, we obtain the Lipschitz property of the value function and the Aubin property of the solution map to a parametric discrete optimal control problem.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Arutyunov, A.V., Marinkovich, B.: Necessary optimality conditions for discrete optimal control problems. Mosc. Univ. Comput. Math. Cybern. 1, 38–44 (2005)
Bertsekas, D.P.: Dynamic Programming and Optimal Control, vol. I. Springer, Berlin (2005)
Gabasov, R., Mordukhovich, B.S., Kirillova, F.M.: The discrete maximum principle. Dokl. Akad. Nauk SSSR. 213, 19–22 (1973). (Russian; English transl. in Soviet Math. Dokl. 14, 1624–1627, 1973)
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)
Ioffe, A.D., Tikhomirov, V.M.: Theory of Extremal Problems. North-Holland Publishing Company, North-Holland (1979)
Larson, R.E., Casti, J.: Principles of Dynamic Programming I. Marcel Dekker, New York (1982)
Larson, R.E., Casti, J.: Principles of Dynamic Programming II. Marcel Dekker, New York (1982)
Lian, Z., Liu, L., Neuts, M.F.: A discrete-time model for common lifetime inventory systems. Math. Oper. Res. 30, 718–732 (2005)
Malanowski, K.: Differential sentivity of solutions of convex constrained optimal control problems for discrete systems. J. Optim. Theory Appl. 53, 429–449 (1987)
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)
Mordukhovich, B.S.: Difference approximations of optimal control system. Prikladaya Mat. Mekhanika 42, 431–440 (1978). (Russian; English transl. in J. Appl. Math. Mech. 42, 452–461 1978)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I, Basis Theory. Springer, Berlin (2006)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation II, Applications. Springer, Berlin (2006)
Mordukhovich, B.S.: Generalized differential calculus for nonsmooth and set-value mappings. J. Math. Anal. Appl. 183, 250–288 (1994)
Nam, N.M.: Coderivatives of normal cone mappings and Lipschitzian stability of parametric variational inequalities. Nonlinear Anal. 73, 2271–2281 (2010)
Pindyck, R.S.: An application of the linear quadratic tracking problem to economic stabilization policy. IEEE Trans. Autom. Control 17, 287–300 (1972)
Qui, N.T.: New results on linearly perturbed polyhedral normal cone mapping. J. Math. Anal. Appl. 381, 352–364 (2011)
Rockafellar, R.T., Wets, R. J.-B.: Variational Analysis. Springer, Berlin (1998)
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)
Thuy, L.Q., Thanh, B.T., Toan, N.T.: On the no-gap second-order optimality conditions for a discrete optimal control problem with mixed constraints. J. Optim. Theory Appl. 173, 421–442 (2017)
Thuy, L.Q., Toan, N.T.: The Mordukhovich coderivative and the local metric regularity of the solution map to a parametric discrete optimal control problem. Acta Math. Vietnam. 43, 175–199 (2018)
Toan, N.T., Kien, B.T.: Continuity properties of the solution map to a parametric discrete optimal control problem. J. Nonlinear Convex Anal. 12, 635–650 (2011)
Toan, N.T., Yao, J.-C.: Mordukhovich subgradients of the value function to a parametric discrete optimal control problem. J. Global Optim. 58, 595–612 (2014)
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)
Toan, N.T., Thuy, L.Q.: Second-order necessary optimality conditions for a discrete optimal control problem with mixed constraints. J. Global Optim. 64, 533–562 (2016)
Tu, P.N.V.: Introductory Optimization Dynamics. Springer, Berlin (1991)
This paper is dedicated to Professor Hoang Tuy
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Toan, N.T., Thuy, L.Q. The Lipschitz Properties of the Value Function and the Solution Map to a Parametric Discrete Optimal Control Problem. Acta Math Vietnam 45, 365–382 (2020). https://doi.org/10.1007/s40306-020-00371-5
- Parametric discrete optimal control problem
- Dynamic programming problem
- Solution map
- Value function
- Lipschitz property
- Aubin property
Mathematics Subject Classification (2010)