convexsetexample

Theorem 1 A necessary and sufficient condition for a set S to be convex is that every convex linear combination in S belongs to S.

In this section, we introduce one of the most important ideas in the theory of optimization, that of a convex set. We discuss other ideas which stem from the ...

convex set: contains line segment between any two points in the set x1, x2 ∈ C, 0 ≤ ≤ 1. =⇒. x1 + (1 − )x2 ∈ C examples (one convex, two nonconvex ...

Convex Sets Examples. Example 1: Prove that C = (x1, x2) : 2x1 + 3x2 = 7} ⊂ R2 is a convex set. Solution: Assume that X, Y ∈ C, where X = (x1, x2), Y = (y ...

Here are some examples of convex sets: Trivial ones: empty set, point, line. Norm ball: x : ||x|| ≤ r}, for given norm || · ||, radius r.

Simple examples of convex sets are: • The empty set ∅, the singleton set x0}, and the complete space Rn;. • Lines aT x = b} ...

1.1.6 Exercise (Examples of convex sets) Prove the following. • A subset A of R is an interval if x, y ∈ A and x<z<y imply z ∈ A. A.