Nondifferentiable Optimization - L. V. Vasil'ev

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Sommaire:
1. Fundamentals of Convex Analysis and Related Problems.- 1. Convex sets. Convex hulls. Separation theorem.- 2. Point-to-set mappings.- 3. Convex cone. Cone of feasible directions. Conjugate cone.- 4. Convex functions. Continuity and directional differentiability.- 5. Subgradients and subdifferentials of convex functions.- 6. Distance from a set to a cone. Conditions for a minimum.- 7. ?-subdifferentials.- 8. Directional ?-derivatives. Continuity of the ?-subdifferential mapping.- 9. Some properties and inequalities for convex functions.- 10. Conditional ?-subdifferentials.- 11. Conditional directional derivatives. Continuity of the conditional ?-subdifferential mapping.- 12. Representation of a convex set by means of inequalities.- 13. Normal cones. Conical mappings.- 14. Directional differentiability of a supremum function.- 15. Differentiability of a convex function.- 16. Conjugate functions.- 17. Computation of ?-subgradients of some classes of convex function.- 2. Quasidifferentiable Functions.- 1. Definition and examples of quasidifferentiable functions.- 2. Basic properties of quasidifferentiable functions. Basic formulas of quasidifferential calculus.- 3. Calculating quasidifferentials: examples.- 4. Quasidifferentiability of convexo-concave functions.- 5. Necessary conditions for an extremum of a quasidifferentiable function on En.- 6. Quasidifferentiable sets.- 7. Necessary conditions for an extremum of a quasidifferentiable function on a quasidifferentiable set.- 8. The distance function from a point to a set.- 9. Implicit function.- 3. Minimization on the Entire Space.- 1. Necessary and sufficient conditions for a minimum of a convex function on En.- 2. Minimization of a smooth function.- 3. The method of steepest descent.- 4. The subgradient methodfor minimizing a convex function.- 5. The multistep subgradient method.- 6. The relaxation-subgradient method.- 7. The relaxation ?-subgradient method.- 8. The Kelley method.- 9. Minimization of a supremum-type function.- 10. Minimization of a convex maximum-type function and the extremum-basis method.- 11. A numerical method for minimizing quasidifferentiable functions.- 4. Constrained Minimization.- 1. Necessary and sufficient conditions for a minimum of a convex function on a convex set.- 2. ?-stationary points.- 3. The conditional gradient method.- 4. The method of steepest descent for the minimization of convex functions.- 5. The (?,?)-subgradient method in the presence of constraints.- 6. The subgradient method with a constant step-size.- 7. The modified (?,?)-subgradient method in the presence of constraints.- 8. The nonsmooth penalty-function method.- 9. The Kelley method for the minimization on a convex set.- 10. The relaxation-subgradient method in the presence of constraints.- Notes and Comments.- References.- Appendix 1. Bibliography and guide to publications on Quasidifferential Calculus.- Appendix 2. Bibliography on Quasidifferential Calculus as of January 1, 1985.- List of forthcoming publications.- Transliteration table.