# Approximation of optimization problems with constraints through kernel Sum-Of-Squares

arXiv: https://arxiv.org/abs/2301.06339 · HTML: https://mlsanity.com/paper/2301.06339/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 42 references that are in the map.
- First posted on arXiv 16 Jan 2023.

## Builds on (most cited first, 11 of 11)

- [Non-parametric Models for Non-negative Functions](https://mlsanity.com/paper/2007.03926/) (2020)
- [Finding Global Minima via Kernel Approximations](https://mlsanity.com/paper/2012.11978/) (2020)
- [Learning Dynamical Systems with Side Information](https://mlsanity.com/paper/2008.10135/) (2020)
- [A Dimension-free Computational Upper-bound for Smooth Optimal Transport Estimation](https://mlsanity.com/paper/2101.05380/) (2021)
- [Hard Shape-Constrained Kernel Machines](https://mlsanity.com/paper/2005.12636/) (2020)
- [Nonparametric Shape-restricted Regression](https://mlsanity.com/paper/1709.05707/) (2017)
- [Sampling from Arbitrary Functions via PSD Models](https://mlsanity.com/paper/2110.10527/) (2021)
- [Infinite-Dimensional Sums-of-Squares for Optimal Control](https://mlsanity.com/paper/2110.07396/) (2021)
- [Learning PSD-valued functions using kernel sums-of-squares](https://mlsanity.com/paper/2111.11306/) (2021)
- [Handling Hard Affine SDP Shape Constraints in RKHSs](https://mlsanity.com/paper/2101.01519/) (2021)
- [Linearly-constrained Linear Quadratic Regulator from the viewpoint of kernel methods](https://mlsanity.com/paper/2011.02196/) (2020)

## Built on by (most cited first, 2 of 2)

- [An Efficient Model Maintenance Approach for MLOps](https://mlsanity.com/paper/2412.04657/) (2024)
- [Scalable and adaptive prediction bands with kernel sum-of-squares](https://mlsanity.com/paper/2505.21039/) (2025)

## Abstract (arXiv)

Handling an infinite number of inequality constraints in infinite-dimensional spaces occurs in many fields, from global optimization to optimal transport. These problems have been tackled individually in several previous articles through kernel Sum-Of-Squares (kSoS) approximations. We propose here a unified theorem to prove convergence guarantees for these schemes. Pointwise inequalities are turned into equalities within a class of nonnegative kSoS functions. Assuming further that the functions appearing in the problem are smooth, focusing on pointwise equality constraints enables the use of scattering inequalities to mitigate the curse of dimensionality in sampling the constraints. Our approach is illustrated in learning vector fields with side information, here the invariance of a set.