On the (Non-)Differentiability of the Optimal Value Function When the Optimal Solution Is Unique

Oyama, Daisuke; Takenawa, Tomoyuki

On the (Non-)Differentiability of the Optimal Value Function When the Optimal Solution Is Unique

Daisuke Oyama
Faculty of Economics, University of Tokyo

and

Tomoyuki Takenawa
Faculty of Marine Technology, Tokyo University of Marine Science and Technology

Abstract

We present examples of a parameterized optimization problem, with a continuous objective function differentiable with respect to the parameter, that admits a unique optimal solution, but whose optimal value function is not differentiable. We also show independence of Danskin's and Milgrom and Segal's envelope theorems. Journal of Economic Literature Classification Number: C61.

Key Words: parameterized optimization; sensitivity analysis; value function; envelope theorem.

Journal of Mathematical Economics 76 (2018), 21-32.
First draft: December 25, 2013; this version: January 30, 2018. PDF file, Supplementary Appendix