# Derivative of a Conic Problem with a Unique Solution

## TL;DR

This paper establishes conditions under which the solution map of a conic optimization problem with a unique solution is differentiable, providing a derivative operator useful for applications like machine learning and control.

## Contribution

It generalizes previous results by deriving the derivative of the solution map for any convex conic problem under regularity conditions, with a practical Python implementation.

## Key findings

- Solution map is differentiable under regularity conditions.
- Derived an abstract linear operator for the derivative.
- Applicable to practical problems with cone constraints.

## Abstract

We view a conic optimization problem that has a unique solution as a map from its data to its solution. If sufficient regularity conditions hold at a solution point, namely that the implicit function theorem applies to the normalized residual function of [Busseti et al., 2018], the problem solution map is differentiable. We obtain the derivative, in the form of an abstract linear operator. This applies to any convex optimization problem in conic form, while a previous result [Amos et al., 2016] studied strictly convex quadratic programs. Such differentiable problems can be used, for example, in machine learning, control, and related areas, as a layer in an end-to-end learning and control procedure, for backpropagation. We accompany this note with a lightweight Python implementation which can handle problems with the cone constraints commonly used in practice.

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Source: https://tomesphere.com/paper/1903.05753