  This paper proposes a theoretical framework of the grey wolf optimizer (GWO)
based on several interesting theoretical findings, involving sampling
distribution, order-1 and order-2 stability, and global convergence analysis.
In the part I of the paper, the characteristics of the sampling distribution of
the new solution and the probabilistic stability of the GWO are carefully
discussed based on the well-known stagnation assumption for simplification
purposes. Firstly, the characteristics of the sampling distribution of the new
solution, mainly related to the shape of the joint probability density function
(PDF), are discussed under the assumption that the original solution before
updating is constant. Then, the assumption that the original solution is
constant is eliminated to perform the sampling distribution analysis, based on
which several characteristics of the new solution are provided, containing the
shape of the joint PDF and central moments of any positive integer order.
Finally, the order-1 and order-2 stability of the GWO under stagnation
assumption is introduced and proved as the inference of conclusions above,
which are all verified by numerical simulations.
