Abstract
As a complement to a recent paper by An and Yen (Appl Anal 94:108–128, 2015) on subdifferentials of the optimal value function in parametric convex programming under inclusion constraints and functional constraints, this paper studies the differential stability of convex optimization problems under a regularity condition of Aubin’s type (Aubin in Optima and equilibria: an introduction to nonlinear analysis. Springer, New York, 1998). By a suitable sum rule for convex subdifferentials, we obtain exact formulas for the subdifferential and singular subdifferential of the optimal value function. Illustrative examples and a detailed comparison of our results with those of the above-mentioned paper are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
An, D.T.V., Yen, N.D.: Differential stability of convex optimization problems under inclusion constraints. Appl. Anal. 94, 108–128 (2015)
Aubin, J.-P.: Optima and Equilibria: An Introduction to Nonlinear Analysis. Springer, New York (1998)
Auslender, A.: Differential stability in nonconvex and nondifferentiable programming. Math. Program. Stud. 10, 29–41 (1979)
Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)
Dien, P.H., Yen, N.D.: On implicit function theorems for set-valued maps and their application to mathematical programming under inclusion constraints. Appl. Math. Optim. 24, 35–54 (1991)
Gauvin, J., Dubeau, F.: Differential properties of the marginal function in mathematical programming. Math. Program. Stud. 19, 101–119 (1982)
Gauvin, J., Dubeau, F.: Some examples and counterexamples for the stability analysis of nonlinear programming problems. Math. Program. Stud. 21, 69–78 (1984)
Gollan, B.: On the marginal function in nonlinear programming. Math. Oper. Res. 9, 208–221 (1984)
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)
Rockafellar, R.T.: Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming. Math. Program. Stud. 17, 28–66 (1982)
Thibault, L.: On subdifferentials of optimal value functions. SIAM J. Control Optim. 29, 1019–1036 (1991)
Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North-Holland Publishing Company, Amsterdam (1979)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. Volume I: Basic Theory, Volume II: Applications. Springer, Berlin (2006)
Kolmogorov, A.N., Fomin, S.V.: Introductory Real Analysis. Dover Publications, New York (1975)
Luenberger, D.G.: Optimization by Vector Space Methods. Wiley, New York (1969)
Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill, New York (1991)
Acknowledgments
The research of Duong Thi Viet An was supported by the National Foundation for Science & Technology Development (Vietnam) under grant number 101.01-2014.37 and College of Sciences, Thai Nguyen University. The research of Jen-Chih Yao was supported by the Grant MOST 102-2221-E-039-017-MY3. The authors thank Prof. Nguyen Dong Yen for useful discussions, Mr. Vu Xuan Truong for pointing us the proof of the geometrical form of the Moreau–Rockafellar theorem in [4] and two anonymous referees for valuable remarks.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
An, D.T.V., Yao, JC. Further Results on Differential Stability of Convex Optimization Problems. J Optim Theory Appl 170, 28–42 (2016). https://doi.org/10.1007/s10957-016-0900-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-016-0900-7
Keywords
- Parametric convex programming
- Optimal value function
- Subdifferential
- Singular subdifferential
- Aubin’s regularity condition