Minimization Methods for Non-Differentiable Functions - Shor, Naum Zuselevich (Springer Berlin Heidelberg, 1985)
In recent years much attention has been given to the development of automatic systems of planning, design and control in various branches of the national economy. Quality of decisions is an issue which has come to the forefront, increasing the significance of optimization algorithms in mathematical software packages for automatic systems of various levels and purposes. Methods for minimizing functions with discontinuous gradients are gaining in importance and the experts in the computational methods of mathematical programming tend to agree that progress in the development of algorithms for minimizing nonsmooth functions is the key to the construction of efficient techniques for solving large scale problems. This monograph summarizes to a certain extent fifteen years of the author's work on developing generalized gradient methods for nonsmooth minimization. This work started in the department of economic cybernetics of the Institute of Cybernetics of the Ukrainian Academy of Sciences under the supervision of V.S. Mikhalevich, a member of the Ukrainian Academy of Sciences, in connection with the need for solutions to important, practical problems of optimal planning and design. In Chap. I we describe basic classes of nonsmooth functions that are differentiable almost everywhere, and analyze various ways of defining generalized gradient sets. In Chap. 2 we study in detail various versions of the subgradient method, show their relation to the methods of Fejer-type approximations and briefly present the fundamentals of e-subgradient methods