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The following topics were substantially covered in the class:

1.  (definitions, geometry, intersections, examples of:)
     Convex sets 
     convex cones (and the dual/polar), separation theorems 
     convex functions (directional derivatives, characterizations, epigraph)
     strong convexity.


2. Convex optimization problems :
      - definition of convex opt problem
      - inequality constrained problems; generalized inequality
      constrained problems
      - Rockafellar-Pshenichni optimality condition

3. Optimality (KKT) and (Lagrangian/weak/strong) Duality
      - how to write down the Lagrangian
      - how to find the hidden constraints
      - how to write down the dual functional
      - how to write down the dual

      - weak duality theorem
      - strong duality theorem
      - constraint qualifications (e.g. Slater condition/when needed),
      conditions that imply strong duality
      - KKT conditions (difference between sufficiency and necessity)

4. Steepest descent algorithm for function minimization
       -  deriving the steepest descent direction
       -  steplength decisions

5. Newton's method
      - Newton's method for system of nonlinear equations
      - how to write the Newton system for an unconstrained /
      equality constrained problem
      - damped Newton's method
      - Newton decrement
      - importance/role of quadratic functions

6. interior point method
      - log-barrier function : find the log-barrier function
      corresponding to an inequality constraint
      - barrier method : writing an inequality-constrained
      problem as an unconstrained problem with the new
      objective function including the barrier function
      - log-barrier problem applied to LP, and derivation of modern primal-dual
        interior point methods for LP


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• An outline of the sections to study from the midterm outline and the later chapters:
  1. 2.1,2.2,2.3 (pages 21-38)
     2.5-2.5.2 (pages 46-51)
  2. 3.1 (page 67)
     3.1.3-3.1.7 (pages 69-77)
     3.2-3.2.4 (pages 79-87)
  3. 4.2.1-4.2.3 (pages 136-142)
     4.3 (pages 146-148)
     4.4 (pages 152-156)
  4. 5.1 (pages 215-220)
     5.2 (pages 223-226)
     5.5 (pages 241-249)
     5.6 (pages 249-258)
  5. 9.1-9.5 (pages 457-496)
  6. 10.1-10.2.3 (pages 521-529)
     10.3.1-10.3.2 (pages 531-536)
  7. 11.1-11.3.2 (pages 561-572)
     11.3.4-11.4.2 (pages 577-582)
     11.7-11.8.1 (pages 609-617)