On the Differentiability of the Solution to Convex Optimization Problems

TL;DR

This paper establishes conditions under which the solution to convex optimization problems is differentiable with respect to problem data, using the implicit function theorem on the KKT system.

Contribution

It provides a rigorous framework for differentiability of convex optimization solutions based on Slater's condition and Jacobian non-singularity.

Findings

Differentiability holds when Slater's condition is satisfied.

Twice differentiability of involved functions is required.

A non-singular Jacobian ensures differentiability of solutions.

Abstract

In this paper, we provide conditions under which one can take derivatives of the solution to convex optimization problems with respect to problem data. These conditions are (roughly) that Slater's condition holds, the functions involved are twice differentiable, and that a certain Jacobian matrix is non-singular. The derivation involves applying the implicit function theorem to the necessary and sufficient KKT system for optimality.