# On The Mathematics of the Natural Physics of Optimization

arXiv: https://arxiv.org/abs/2604.17645 · HTML: https://mlsanity.com/paper/2604.17645/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 81 references that are in the map.
- First posted on arXiv 19 Apr 2026.

## Builds on (most cited first, 14 of 14)

- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [A Variational Perspective on Accelerated Methods in Optimization](https://mlsanity.com/paper/1603.04245/) (2016)
- [Analysis and Design of Optimization Algorithms via Integral Quadratic Constraints](https://mlsanity.com/paper/1408.3595/) (2014)
- [Understanding the Acceleration Phenomenon via High-Resolution Differential Equations](https://mlsanity.com/paper/1810.08907/) (2018)
- [Coordinate Descent Converges Faster with the Gauss-Southwell Rule Than Random Selection](https://mlsanity.com/paper/1506.00552/) (2015)
- [The Approximate Duality Gap Technique: A Unified Theory of First-Order Methods](https://mlsanity.com/paper/1712.02485/) (2017)
- [On dissipative symplectic integration with applications to gradient-based optimization](https://mlsanity.com/paper/2004.06840/) (2020)
- [Generalized Momentum-Based Methods: A Hamiltonian Perspective](https://mlsanity.com/paper/1906.00436/) (2019)
- [An Optimal Control Theory for Accelerated Optimization](https://mlsanity.com/paper/1902.09004/) (2019)
- [Global Dynamical Solvers for Nonlinear Programming Problems](https://mlsanity.com/paper/1512.06955/) (2015)
- [An Arrow-Hurwicz-Uzawa Type Flow as Least Squares Solver for Network Linear Equations](https://mlsanity.com/paper/1701.03908/) (2017)
- [Nonsmooth stabilization and its computational aspects](https://mlsanity.com/paper/2006.14013/) (2020)
- [Derivation of Coordinate Descent Algorithms from Optimal Control Theory](https://mlsanity.com/paper/2309.03990/) (2023)

## Abstract (arXiv)

A number of optimization algorithms have been inspired by the physics of Newtonian motion. Here, we ask the question: do algorithms themselves obey some ``natural laws of motion,'' and can they be derived by an application of these laws? We explore this question by positing the theory that optimization algorithms may be considered as some manifestation of hidden algorithm primitives that obey certain universal non-Newtonian dynamics. This natural physics of optimization is developed by equating the terminal transversality conditions of an optimal control problem to the generalized Karush/John-Kuhn-Tucker conditions of an optimization problem. Through this equivalence formulation, the data functions of a given constrained optimization problem generate a natural vector field that permeates an entire hidden space with information on the optimality conditions. An ``action-at-a-distance'' operation via a Pontryagin-type minimum principle produces a local action to deliver a globalized result by way of a Hamilton-Jacobi inequality. An inverse-optimal algorithm is generated by performing control jumps that dissipate quantized ``energy'' defined by a search Lyapunov function. Illustrative applications of the proposed theory show that a large number of algorithms can be generated and explained in terms of the new mathematical physics of optimization.