Generalized Envelope Theorems: Applications to Dynamic Programming

Olivier Morand, Kevin Reffett, Suchismita Tarafdar

Research output: Contribution to journalArticle

Abstract

We show in this paper that the class of Lipschitz functions provides a suitable framework for the generalization of classical envelope theorems for a broad class of constrained programs relevant to economic models, in which nonconvexities play a key role, and where the primitives may not be continuously differentiable. We give sufficient conditions for the value function of a Lipschitz program to inherit the Lipschitz property and obtain bounds for its upper and lower directional Dini derivatives. With strengthened assumptions we derive sufficient conditions for the directional differentiability, Clarke regularity, and differentiability of the value function, thus obtaining a collection of generalized envelope theorems encompassing many existing results in the literature. Some of our findings are then applied to decision models with discrete choices, to dynamic programming with and without concavity, to the problem of existence and characterization of Markov equilibrium in dynamic economies with nonconvexities, and to show the existence of monotone controls in constrained lattice programming problems.

Original languageEnglish (US)
Pages (from-to)1-38
Number of pages38
JournalJournal of Optimization Theory and Applications
DOIs
StateAccepted/In press - Feb 22 2018

Keywords

  • Clarke regularity
  • Envelope theorem
  • Lipschitz dynamic programming
  • Lipschitz function
  • Recursive dynamic program
  • Supermodularity

ASJC Scopus subject areas

  • Control and Optimization
  • Management Science and Operations Research
  • Applied Mathematics

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