CMPS 290c Topics - Spring ’08
1. Convex Sets
•Affine sets
•Convex sets
•Convex combinations and convex hull
•cones and conic combinations
•hyperplanes and half-spaces
•Norms and Norm balls
•polyhedra
•Convexity preserving operations: intersection, image of convex
set under affine transformation
•Perspective function
•dual cones, generalized (conic) inequalities, minimum/minimal
elements
•Separating hyperplane theorem
•Supporting hyperplane theorem
2. Convex functions
•Definitions of convexity, concavity, strict convexity
•convex if every restriction to a line is convex
•First order condition
•Second order condition
•Epigraphs and sub-level sets, Jensen’s inequality
•Closure under: non-negative weighted sums, composition with
affine functions, point-wise max and sup, composition with scalar
function, minimization over convex set
•conjugate function
•quasi-convexity
3. Optimization problems
•Problems in standard form
•Domain; feasible, optimal, and locally optimal points
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