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Full Description
In recent years much attention has been given to the development of auto matic 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 math ematical software packages for al,ltomatic systems of various levels and pur poses. Methods for minimizing functions with discontinuous gradients are gaining in importance and the ‾xperts 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 con struction 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 dif ferentiable almost everywhere, and analyze various ways of defining generalized gradient sets. In Chap. 2 we study in detail various versions of the su bgradient method, show their relation to the methods of Fejer-type approximations and briefly present the fundamentals of e-subgradient methods.
Contents
1. Special Classes of Nondifferentiable Functions and Generalizations of the Concept of the Gradient.- 1.1 The Need to Introduce Special Classes of Nondifferentiable Functions.- 1.2 Convex Functions. The Concept of Subgradient.- 1.3 Some Methods for Computing Subgradients.- 1.4 Almost Differentiable Functions.- 1.5 Semismooth and Semiconvex Functions.- 2. The Subgradient Method.- 2.1 The Problem of Stepsize Selection in the Subgradient Method.- 2.2 Basic Convergence Results for the Subgradient Method.- 2.3 On the Linear Rate of Convergence of the Subgradient Method.- 2.4 The Subgradient Method and Fejer-type Approximations.- 2.5 Methods of ?-subgradients.- 2.6 An Extension of the Subgradient Method to a Class of Nonconvex Functions. Stochastic Versions and Stability of the Method.- 3. Gradient-type Methods with Space Dilation.- 3.1 Heuristics of Methods with Space Dilation.- 3.2 Operators of Space Dilation.- 3.3 The Subgradient Method with Space Dilation in the Direction of the Gradient.- 3.4 Convergence of Algorithms with Space Dilation.- 3.5 Application of the Subgradient Method with Space Dilation to the Solution of Systems of Nonlinear Equations.- 3.6 A Minimization Method Using the Operation of Space Dilation in the Direction of the Difference of Two Successive Almost-Gradients.- 3.7 Convergence of a Version of the r-Algorithm with Exact Directional Minimization.- 3.8 Relations between SDG Algorithms and Algorithms of Successive Sections.- 3.9 Computational Modifications of Subgradient Methods with Space Dilation.- 4. Applications of Methods for Nonsmooth Optimization to the Solution of Mathematical Programming Problems.- 4.1 Application of Subgradient Algorithms in Decomposition Methods.- 4.2 An Iterative Method for Solving Linear Programming Problems of SpecialStructure.- 4.3 The Solution of Distribution Problems by the Subgradient Method.- 4.4 Experience in Solving Production-Transportation Problems by Subgradient Algorithms with Space Dilation.- 4.5 Application of r-Algorithms to Nonlinear Minimax Problems.- 4.6 Application of Methods for Minimizing Nonsmooth Functions to Problems of Interpreting Gravimetric Observations.- 4.7 Other Areas of Applications of Generalized Gradient Methods.- Concluding Remarks.- References.
