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An easy path to convex analysis and applications /

Bibliographic Details
Main Authors: Mordukhovich, B. Sh. (Boris Sholimovich) (Author), Nam, Nguyen Mau (Author)
Format: eBook
Language:English
Published: San Rafael, California (1537 Fourth Street, San Rafael, CA 94901 USA) : Morgan & Claypool, 2014.
Series:Synthesis lectures on mathematics and statistics ; # 14.
Subjects:
Convex geometry.
Affine set
Carathéodory theorem
Fenchel conjugate
Fermat-Torricelli problem
Helly theorem
Nash equilibrium
Radon theorem
Weiszfeld algorithm
convex function
convex set
directional derivative
distance function
generalized differentiation
minimal time function
normal cone
optimal value function
optimization
set-valued mapping
smallest enclosing ball problem
subdifferential
subgradient
subgradient algorithm
support function
Electronic books.
Online Access:Connect to the full text of this electronic book
Table of Contents:
  • 1. Convex sets and functions
  • 1.1 Preliminaries
  • 1.2 Convex sets
  • 1.3 Convex functions
  • 1.4 Relative interiors of convex sets
  • 1.5 The distance function
  • 1.6 Exercises for chapter 1
  • 2. Subdifferential calculus
  • 2.1 Convex separation
  • 2.2 Normals to convex sets
  • 2.3 Lipschitz continuity of convex functions
  • 2.4 Subgradients of convex functions
  • 2.5 Basic calculus rules
  • 2.6 Subgradients of optimal value functions
  • 2.7 Subgradients of support functions
  • 2.8 Fenchel conjugates
  • 2.9 Directional derivatives
  • 2.10 Subgradients of supremum functions
  • 2.11 Exercises for chapter 2
  • 3. Remarkable consequences of convexity
  • 3.1 Characterizations of differentiability
  • 3.2 Carathéodory theorem and Farkas Lemma
  • 3.3 Radon theorem and Helly theorem
  • 3.4 Tangents to convex sets
  • 3.5 Mean value theorems
  • 3.6 Horizon cones
  • 3.7 Minimal time functions and Minkowski gauge
  • 3.8 Subgradients of minimal time functions
  • 3.9 Nash equilibrium
  • 3.10 Exercises for chapter 3
  • 4. Applications to optimization and location problems
  • 4.1 Lower semicontinuity and existence of minimizers
  • 4.2 Optimality conditions
  • 4.3 Subgradient methods in convex optimization
  • 4.4 The Fermat-Torricelli problem
  • 4.5 A generalized Fermat-Torricelli problem
  • 4.6 A generalized Sylvester problem
  • 4.7 Exercises for chapter 4
  • Solutions and hints for exercises
  • Bibliography
  • Authors' biographies
  • Index.