  Poisson restart assumes that a stochastic process is interrupted and starts
again at random time moments. A number of studies have demonstrated that this
strategy may minimize the expected completion time in some classes of random
search tasks. What is more, it turned out that under optimally tuned restart
rate, any stochastic process, regardless of its nature and statistical details,
satisfies a number of universal relations for the statistical moments of
completion time. In this paper, we describe several new universal properties of
optimally restarted processes. Also we obtain a universal inequality for the
quadratic statistical moments of completion time in the optimization problem
where stochastic process has several possible completion scenarios.
