Convex Set
Convex Optimization Solutions Manual
Stephen Boyd
Lieven Vandenberghe
January 4, 2006
Chapter 2
Convex sets
Exercises
Exercises
Definition of convexity
2.1 Let C ⊆ Rn be a convex set, with x1 , . . . , xk ∈ C, and let θ1 , . . . , θk ∈ R satisfy θi ≥ 0, θ1 + · · · + θk = 1. Show that θ1 x1 + · · · + θk xk ∈ C. (The definition of convexity is that this holds for k = 2; you must show it for arbitrary k.) Hint. Use induction on k. Solution. This is readily shown by induction from the definition of convex set. We illustrate the idea for k = 3, leaving the general case to the reader. Suppose that x 1 , x2 , x3 ∈ C, and θ1 + θ2 + θ3 = 1 with θ1 , θ2 , θ3 ≥ 0. We will show that y = θ1 x1 + θ2 x2 + θ3 x3 ∈ C. At least one of the θi is not equal to one; without loss of generality we can assume that θ1 = 1. Then we can write where µ2 = θ2 /(1 − θ1 ) and µ2 = θ3 /(1 − θ1 ). Note that µ2 , µ3 ≥ 0 and µ1 + µ 2 = y = θ1 x1 + (1 − θ1 )(µ2 x2 + µ3 x3 )
1 − θ1 θ2 + θ 3 = = 1. 1 − θ1 1 − θ1 Since C is convex and x2 , x3 ∈ C, we conclude that µ2 x2 + µ3 x3 ∈ C. Since this point and x1 are in C, y ∈ C. 2.2 Show that a set is convex if and only if its intersection with any line is convex. Show that a set is affine if and only if its intersection with any line is affine. Solution. We prove the first part. The intersection of two convex sets is convex. Therefore if S is a convex set, the intersection of S with a line is convex. Conversely, suppose the intersection of S with any line is convex. Take any two distinct points x1 and x2 ∈ S. The intersection of S with the line through x1 and x2 is convex. Therefore convex combinations of x1 and x2 belong to the intersection, hence also to S. 2.3 Midpoint convexity. A set C is midpoint convex if whenever two points a, b are in C, the average or midpoint (a + b)/2 is in C. Obviously a convex set is midpoint convex. It can be proved that under mild conditions midpoint convexity implies convexity. As a simple case, prove that
Stephen Boyd
Lieven Vandenberghe
January 4, 2006
Chapter 2
Convex sets
Exercises
Exercises
Definition of convexity
2.1 Let C ⊆ Rn be a convex set, with x1 , . . . , xk ∈ C, and let θ1 , . . . , θk ∈ R satisfy θi ≥ 0, θ1 + · · · + θk = 1. Show that θ1 x1 + · · · + θk xk ∈ C. (The definition of convexity is that this holds for k = 2; you must show it for arbitrary k.) Hint. Use induction on k. Solution. This is readily shown by induction from the definition of convex set. We illustrate the idea for k = 3, leaving the general case to the reader. Suppose that x 1 , x2 , x3 ∈ C, and θ1 + θ2 + θ3 = 1 with θ1 , θ2 , θ3 ≥ 0. We will show that y = θ1 x1 + θ2 x2 + θ3 x3 ∈ C. At least one of the θi is not equal to one; without loss of generality we can assume that θ1 = 1. Then we can write where µ2 = θ2 /(1 − θ1 ) and µ2 = θ3 /(1 − θ1 ). Note that µ2 , µ3 ≥ 0 and µ1 + µ 2 = y = θ1 x1 + (1 − θ1 )(µ2 x2 + µ3 x3 )
1 − θ1 θ2 + θ 3 = = 1. 1 − θ1 1 − θ1 Since C is convex and x2 , x3 ∈ C, we conclude that µ2 x2 + µ3 x3 ∈ C. Since this point and x1 are in C, y ∈ C. 2.2 Show that a set is convex if and only if its intersection with any line is convex. Show that a set is affine if and only if its intersection with any line is affine. Solution. We prove the first part. The intersection of two convex sets is convex. Therefore if S is a convex set, the intersection of S with a line is convex. Conversely, suppose the intersection of S with any line is convex. Take any two distinct points x1 and x2 ∈ S. The intersection of S with the line through x1 and x2 is convex. Therefore convex combinations of x1 and x2 belong to the intersection, hence also to S. 2.3 Midpoint convexity. A set C is midpoint convex if whenever two points a, b are in C, the average or midpoint (a + b)/2 is in C. Obviously a convex set is midpoint convex. It can be proved that under mild conditions midpoint convexity implies convexity. As a simple case, prove that