Stochastic generalized gradient method for nonconvex nonsmooth stochastic optimization

Catastrophe risk management for sustainable development of regions under risks of natural disasters

“…A formal analysis of this method, in particular, for discontinuous objective functions, is based on the general ideas of the stochastic quasigradient method (see [10,20] and further references in [12,15,16,19]). …”

Some scientific results of Yu. M. Ermoliev and his school in modern stochastic optimization theory

“…One more possible way to get rid of local solutions is sequential smooth approximation [12]. Other global optimization methods are considered in [13]. Example 5.…”

“…The SQG method allows obtaining simple finite-difference approximations for a nonsmooth optimization problem in the general case (both deterministic and stochastic). Small randomization of (7)-(10) due to the replacement of a given point x k with a random point x x k k k = +n , where the random vector n k has density and || || n k ® 0 with probability 1 guarantees their convergence for locally Lipschitzian and discontinuous functions [1, [11][12][13][14][15]. Assume that F x ( ) is a locally integrable (probably discontinuous) function and the vector n k has sufficiently smooth density concentrated on a bounded set.…”

“…If, along with the observations of stochastic parameters, the algorithm includes other stochastic elements (such as search randomization), they should also be included in the structure of the space ( , , ) W S w w P . (ii) In the general case, directions x( ) k are estimates of the subgradients of function F x ( ) at the point x k and should [11][12][13][14][15] can be used to determine how the choice of variable x influences the system as a whole. Any such modeling can be considered as observation of a "medium" w from a sample space W .…”

“…, and X is a convex compact set, then { } F x k ( ) converges with probability 1, and limiting points of the random sequence { } x k belong with probability 1 to the connected set of local solutions [12,13]. The convergence of (21) for e k º 0 holds for so-called [1, 14,15] weakly…”