Schedule

Week 1

Monday 8/28/2017

• introduction to the course, policies, etc.
• description of lines and hyperplanes
• combinations: linear, affine, conic, convex

Wednesday 8/30/2017

• three representations of hyperplanes
• graphical solution to linear programming problem
• combinations: linear, affine, conic, convex

Friday 9/1/2017

• affine hull, conic hull, convex hull
• affine sets, cones, convex cones, convex sets
• interior and relative interior of a set
• convex objects: hyperplane, subspace, halfspace, slab, norm ball

Week 2

Wednesday 9/6/2017

• operations that preserve convexity: intersections; images and inverse images under affine transformations, perspective transformations, linear fractional functions; and projection onto some coordinates
• inverse image examples
• separating hyperplane theorem

Friday 9/8/2017

• applications of the separating hyperplane theorem
• theorems of the alternative
• supporting hyperplanes
• supporting hyperplane theorem

Week 3

• Watch Boyd's lecture 3 (convex functions) online

Friday 9/15/2017

• Taylor approximation of differentiable functions
• Summary of convex functions
• Definition of convex functions

Week 4

Monday 9/18/2017

• First order condition for convexity
• Second order condition for convexity
• Examples of convex functions

Wednesday 9/20/2017

• Sublevel sets and convexity
• Graphs, epigraphs, hypographs, level sets
• Supporting hyperplane theorem and epigraphs of convex functions
• Jensen's inequality
• Maximization and convexity

Friday 9/22/2017

• Compositions and examples
• Minimization and examples
• Perspective function
• Introduction to conjugate functions

Week 5

Monday 9/25/2017

• Conjugate function definition and interpretation
• Procedure for computing conjugage functions
• Examples of conjugate functions

Wednesday 9/27/2017

Friday 9/29/2017

• Convexit with respect to generalized inequalities
• Convexity on a line, first order condition
• Introduction to constrained optimization problems
• Terminology (optimum value, optimum point, active/inactive constraints, unbounded below, unattained, etc.)