Derivative of a Conic Problem with a Unique Solution

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.