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Gradient methods for nonlocal optimization using potential theory. (English. Russian original) Zbl 0837.90109
Phys.-Dokl. 38, No. 5, 178-180 (1993); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 330, No. 2, 170-172 (1993).
A new approach for unconstrained optimization of nonconvex nonsmooth functions is suggested. The approach is based on replacing the original problem of minimizing \(f\) in \(\mathbb{R}^n\) by the equivalent randomized problem of minimizing the functional \(\int_{\mathbb{R}^n} f(x) p(x)dx\) over all density functions \(p\). The variation of the randomized functional along a random vector \(Y\) is given. Maximization of this variation over all \(Y\) determines the gradient direction. This direction is characterized by a mean vector field. To make this characterization constructive it is necessary to partition the mean vector field in a scalar potential field and a nondivergent vector field. The corresponding potential function \(\phi\) satisfies the Poisson equation \(\Delta\phi(x)= -(f(x)- c)q(x)\), where \(q\) is a controlled density function, \(c\) is the mean value of \(f\) over \(q\). The potential \(\phi\) serves as a model of extremal properties of \(f\). The first and the second order methods (with respect to \(\phi\)) can be developed for minimizing nondifferentiable nonconvex functions \(f\) [see the authors, Autom. Remote Control 54, No 7, pt. 1, 1077-1086 (1993; Zbl 0830.90126)]. The potential can be used as a Lyapunov function for convergence analysis of these methods [Autom. Remote Control No. 9-11 (1994)].
Reviewer: A.Propoj (Moskva)
90C30 Nonlinear programming
49J52 Nonsmooth analysis