convex sets

(7 hours to learn)


A set S in R^d is convex if for any two points x and y in S, the line segment connecting x and y is also contained in S. Convex sets are part of the definition of convex optimization problems, a very general class of optimization problems for which the optimal solution can often be found.


This concept has the prerequisites:


  • Know how convex sets are defined, both
    • geometrically (line segments lie within the set)
    • algebraically (in terms of linear combinations of vectors)
  • Know some common examples of convex sets (e.g. hyperplanes, halfspaces, Euclidean balls)

Core resources (read/watch one of the following)


Convex Optimization
A graduate-level textbook on convex optimization.
Authors: Stephen Boyd,Lieven Vandenberghe
Other notes:
  • Some of the examples may involve math you haven't seen; if so, just try to get the gist.

See also