Differential stability of convex optimization problems under inclusion constraints

Abstract

Motivated by the recent work of Mordukhovich et al. [Subgradients of marginal functions in parametric mathematical programming. Math. Program. Ser. B. 2009;116:369–396] on the optimal value function in parametric programming under inclusion constraints, this paper presents some new results on differential stability of convex optimization problems under inclusion constraints and functional constraints in Hausdorff locally convex topological vector spaces. By using the Moreau–Rockafellar theorem and appropriate regularity conditions, we obtain formulas for computing the subdifferential and the singular subdifferential of the optimal value function. By virtue of the convexity, several assumptions used in the above paper by Mordukhovich et al., like the nonemptyness of the Fréchet upper subdiffential of the objective function, the existence of a local upper Lipschitzian selection of the solution map, as well as the -inner semicontinuity and the -inner semicompactness of the solution map, are no longer needed. Relationships between our results and the corresponding ones in Aubin’s book [Optima and equilibria. An introduction to nonlinear analysis. 2nd ed. New York (NY): Springer; 1998] are discussed.

Keywords:

• parametric programming under inclusion constraints
• convexity
• optimal value function
• subdifferential
• singular subdifferential
• the Moreau–Rockafellar theorem
• normal cone to the sublevel set of a convex function

AMS Subject Classifications:

• 49J53
• 49Q12
• 90C25
• 90C31

Acknowledgments

Very useful comments of Professor Le Dung Muu and the anonymous referee on earlier versions of this paper are gratefully acknowledged.

Notes

Dedicated to Professor Boris Sholimovich Mordukhovich on the occasion of his sixty-fifth birthday.

2 Supplemental data for this article can be accessed http://dx.doi.org/10.1080/00036811.2014.890710.

The research of Duong Thi Viet An was supported by College of Sciences, Thai Nguyen University. The research of Nguyen Dong Yen was supported by the National Foundation for Science & Technology Development (Vietnam) under [grant number 101.02-2011.01].