On the Differentiability of the Solution to Convex Optimization Problems

By Shane T. Barratt. 2018

62 citations recorded by the source corpus.

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.

Subfields: Civil and Structural Engineering, Computational Theory and Mathematics, Numerical Analysis, Mathematical Optimization

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Civil and Structural Engineering

1. Chi Zhang
2. Chunhua Shen
3. Yujun Cai
4. Guosheng Lin
5. Stefano Soatto
6. Avinash Ravichandran
7. Kwonjoon Lee
8. Subhransu Maji

In Computational Theory and Mathematics

1. Steven Diamond
2. Stephen P. Boyd
3. Akshay Agrawal
4. Chi Zhang
5. Chunhua Shen
6. Yujun Cai
7. Guosheng Lin
8. J. Kolter

In Numerical Analysis

1. Chi Zhang
2. Chunhua Shen
3. Yujun Cai
4. Guosheng Lin
5. Steven Diamond
6. Shane T. Barratt
7. Stephen P. Boyd
8. J. Kolter

In Mathematical Optimization

1. Chi Zhang
2. Chunhua Shen
3. Yujun Cai
4. Guosheng Lin
5. Shane T. Barratt
6. Subhransu Maji
7. Stefano Soatto
8. Avinash Ravichandran

Citations in the corpus

• Meta-Learning With Differentiable Convex Optimization (2019, methodology): “Theorem 1 (From Barratt [2]) Suppose g(θ̃, λ̃, ν̃, z) = 0.”
• Partially Does It: Towards Scene-Level FG-SBIR with Partial Input (2022, background): “Optimal Transport: In classical transportation problem,m suppliers S = {si|i = 1, . . . ,m} are required to supply n demanders D = {dj |j = 1, . . . , n}.”
• Differentiable Convex Optimization Layers (2019, background): “Recent work has shown how to differentiate through specific subclasses of convex optimization problems, which can be viewed as functions mapping problem data to solutions [6, 31, 10, 1, 4].”
• DeepEMD: Differentiable Earth Mover's Distance for Few-Shot Learning (2020, methodology): “As is indicated by [13], we can apply the implicit function theorem [11], [12], [13] on the optimality (KKT) conditions to obtain the Jacobian.”
• DeepEMD: Few-Shot Image Classification With Differentiable Earth Mover’s Distance and Structured Classifiers (2020, methodology): “Theorem 1 (From Barratt [3]) Suppose g ( θ, ˜ λ, ˜ ν, ˜ x ) = 0 .”