This paper by Paul Milgrom and Ilya Segal extends traditional envelope theorems to optimization problems with arbitrary choice sets, relaxing the convex and topological structure requirements. The authors show that the traditional envelope formula holds at any differentiability point of the value function and provide conditions for the value function to be absolutely continuous, left- and right-differentiable, or fully differentiable. These results are applied to various economic contexts, including mechanism design, convex programming, continuous optimization, saddle-point problems, parameterized constraints, and optimal stopping problems. The paper contributes to a deeper understanding of the structure of economic optimization models by demonstrating that the value function's behavior can be analyzed without relying on specific assumptions about the choice set's structure.This paper by Paul Milgrom and Ilya Segal extends traditional envelope theorems to optimization problems with arbitrary choice sets, relaxing the convex and topological structure requirements. The authors show that the traditional envelope formula holds at any differentiability point of the value function and provide conditions for the value function to be absolutely continuous, left- and right-differentiable, or fully differentiable. These results are applied to various economic contexts, including mechanism design, convex programming, continuous optimization, saddle-point problems, parameterized constraints, and optimal stopping problems. The paper contributes to a deeper understanding of the structure of economic optimization models by demonstrating that the value function's behavior can be analyzed without relying on specific assumptions about the choice set's structure.