Analysis of search optimization methods based on potential theory. II: On the differentiability of the Lyapunov function (Q1920377)

Employing a relationship between a potential of the Newtonian vector field (generated by the objective function) and a Lyapunov function, the authors, first, define the random process of Part I [Autom. Remote Control 55, No. 9, 1316-1323 (1994); see the review above] in discrete time \(N=1,2,\dots\), \[ X^{N+1}=X^N+\alpha_NY^{N+1}, \qquad \alpha_N>0\text{ constant}. \] Furthermore, they introduce assumptions under which the trajectories of the random process \(X^N\) leave the domain \[ D(\varepsilon)= \{x\in\mathbb{R}^n\mid K>\widehat{f}(x)> \varepsilon\} \] (\(K,\varepsilon\) constants, \(+\infty>K> \varepsilon>0\); \(\widehat f=f-c_N\), \(c_N\) constant) almost surely in a finite time. The presented results are proven in the paper.