Minimization Methods for Non-Differentiable Functions

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.