Optimization with delay-induced bifurcations

Abstract

Optimization is finding the best solution, which mathematically amounts to locating the global minimum of some cost function. Optimization is traditionally automated with digital or quantum computers, each having their limitations and none guaranteeing an optimal solution. Here, we conceive a principle behind optimization based on delay-induced bifurcations, which is potentially implementable in non-quantum analog devices. Often, optimization techniques are interpreted via a particle moving in multi-well energy landscape and to prevent confinement to a non-global minima they should incorporate mechanisms to overcome barriers between the minima. Particularly, simulated annealing digitally emulates pushing a fictitious particle over a barrier by random noise, whereas quantum computers utilize tunneling through barriers. In our principle, the barriers are effectively destroyed by delay-induced bifurcations. Although bifurcation scenarios in nonlinear delay-differential equations can be very complex and are notoriously difficult to predict, we hypothesize, verify, and utilize the finding that they become considerably more predictable in dynamical systems, where the right-hand side depends only on the delayed variable and represents a gradient of some potential energy function. By tuning the delay introduced into the gradient descent setting, thanks to global bifurcations destroying local attractors, one could force the system to spontaneously wander around all minima. This would be similar to noise-induced behavior in simulated annealing but achieved deterministically. Ideally, a slow increase and then decrease of the delay should automatically push the system toward the global minimum. We explore the possibility of this scenario and formulate some prerequisites.