Stochastic Dynamics and Combinatorial Optimization

TL;DR

This paper explores the N-phase stochastic dynamical systems, which exhibit properties of both integrable and chaotic dynamics, proposing their use as highly efficient optimizers for complex search and combinatorial problems.

Contribution

The paper introduces a novel optimization method based on N-phase stochastic dynamics, combining features of simulated and chaotic annealing for improved problem-solving efficiency.

Findings

N-phase systems inherit properties of both integrable and chaotic dynamics.

N-phase dynamics naturally facilitate efficient search for optimal solutions.

Proposed method can lead to advanced hardware for combinatorial optimization.

Abstract

Natural dynamics is often dominated by sudden nonlinear processes such as neuroavalanches, gamma-ray bursts, solar flares \emph{etc}. that exhibit scale-free statistics much in the spirit of the logarithmic Ritcher scale for earthquake magnitudes. On phase diagrams, stochastic dynamical systems (DSs) exhibiting this type of dynamics belong to the finite-width phase (N-phase for brevity) that precedes ordinary chaotic behavior and that is known under such names as noise-induced chaos, self-organized criticality, dynamical complexity \emph{etc.} Within the recently formulated approximation-free supersymemtric theory of stochastics, the N-phase can be roughly interpreted as the noise-induced "overlap" between integrable and chaotic deterministic dynamics. As a result, the N-phase dynamics inherits the properties of the both. Here, we analyze this unique set of properties and conclude that the N-phase DSs must naturally be the most efficient optimizers: on one hand, N-phase DSs have integrable flows with well-defined attractors that can be associated with candidate solutions and, on the other hand, the noise-induced attractor-to-attractor dynamics in the N-phase is effectively chaotic or a-periodic so that a DS must avoid revisiting solutions/attractors thus accelerating the search for the best solution. Based on this understanding, we propose a method for stochastic dynamical optimization using the N-phase DSs. This method can be viewed as a hybrid of the simulated and chaotic annealing methods. Our proposition can result in a new generation of hardware devices for efficient solution of various search and/or combinatorial optimization problems.