Crash course

First part: Elements of convex analysis (Nov. 2014 – Jan. 2015)

  • Convex sets (projection, separation, relative interior, cones)
  • Functions (convex functions, subdifferential, semicontinuity, minimizers)
  • Fenchel-Legendre transform (Fenchel-Young inequality, Fenchel-Moreau theorem)
  • Parametric duality (primal and dual problems, qualification conditions, Lagrangian)
  • Fenchel-Rockafellar duality (infimal convolution, Attouch-Brezis theorem, subdifferential calculus)
  • Lagrangian duality (constrained optimization problems, Karush-Kuhn Tucker conditions, Slater condition, complements on cones)

Second part: Algorithms (Jan. 2015 – March 2015)

  • Alpha-averaged operators
  • Proximal point algorithm, Krasnosel’skii-Mann iterations
  • Cocoercive operators, forward backward algorithm
  • Primal dual algorithms: Vu-Condat, Chambolles-Pock, ADMM