  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.
