🤖 AI Summary
This work proposes a novel paradigm termed “Optimized Natural Physics” to investigate whether optimization algorithms adhere to natural laws of motion induced by the objective function. By establishing equivalence between optimal control problems and generalized KKT conditions, the authors construct a natural vector field governed by non-Newtonian dynamics. Leveraging Pontryagin’s minimum principle, Hamilton–Jacobi inequalities, and energy dissipation mechanisms, they design control strategies possessing inverse optimality. This framework not only unifies the interpretation of diverse existing optimization algorithms but also enables the systematic derivation of new ones. The approach demonstrates that global optimization can be achieved through deliberate modulation of jumps and dissipation, thereby providing a physically intuitive and mathematically unified foundation for optimization theory.
📝 Abstract
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.