Analysis of search methods of optimization based on potential theory. III: Convergence of methods (Q1922451)

A deterministic optimization problem of minimizing a nonsmooth multiextremal function \(f:\mathbb{R}^n\to\mathbb{R}\) is considered. Evidently, this problem can be (from the numerical point of view) very complicated. The authors replace the original deterministic problem by a stochastic optimization one of minimizing \(Ef(X)\), where \(X\) is an \(n\)-dimensional random vector; \(E\) denotes the operator of mathematical expectation. Employing a relationship between a potential of the Newtonian vector field (generated by the objective function) and a Lyapunov function, they define the random process at discrete time \(N=1,2,\dots\), \[ X^{N+1}=X^N+ \alpha_NY^{N+1}, \qquad \alpha_N>0 \text{ constant}. \] The aim of the paper is to introduce assumptions under which the random process \(X^N\) converges (in the sense of strong convergence) to a domain of a local extremum of the function \(f(\cdot)\). This third part completes a series of papers [see Zbl 0857.93101 and Zbl 0857.93102] on similar topics. However, while the other papers deal with the problem of leaving 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, this paper goes deeper into the investigation of the approach to the original optimization problem introduced above.