Lecture 6 - Convex Sets

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1 Lecture 6 - Convex Sets Definition A set C R n is called convex if for any x, y C and λ [0, 1], the point λx + (1 λ)y belongs to C. The above definition is equivalent to saying that for any x, y C, the line segment [x, y] is also in C. convex sets nonconvex sets Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 1 / 32

2 Examples of Convex Sets Lines: A line in R n is a set of the form where z, d R n and d 0. [x, y], (x, y) for x, y R n (x y)., R n. A hyperplane is a set of the form L = {z + td : t R}, H = {x R n : a T x = b} (a R n \{0}, b R) The associated half-space is the set H = {x R n : a T x b} Both hyperplanes and half-spaces are convex sets. Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 2 / 32

3 Convexity of Balls Lemma. Let c R n and r > 0. Then the open ball and the closed ball are convex. B(c, r) = {x R n : x c < r} B[c, r] = {x R n : x c r} Note that the norm is an arbitrary norm defined over R n. Proof. In class Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 3 / 32

4 l 1, l 2 and l balls l 1 l l Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 4 / 32

5 Convexity of Ellipsoids An ellipsoid is a set of the form E = {x R n : x T Qx + 2b T x + c 0}, where Q R n n is positive semidefinite, b R n and c R. Proof. Lemma: E is convex. Write E as E = {x R n : f (x) 0} where f (x) x T Qx + 2b T x + c. Take x, y E and λ [0, 1]. Then f (x) 0, f (y) 0. The vector z = λx + (1 λ)y satisfies z T Qz = λ 2 x T Qx + (1 λ) 2 y T Qy + 2λ(1 λ)x T Qy. x T Qy Q 1/2 x Q 1/2 y = x T Qx y T Qy 1 2 (xt Qx + y T Qy) z T Qz λx T Qx + (1 λ)y T Qy f (z) = z T Qz + 2b T z + c λx T Qx + (1 λ)y T Qy + 2λb T x + 2(1 λ)b T y + λc + (1 λ)c = λf (x) + (1 λ)f (y) 0, Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 5 / 32

6 Algebraic Operations Preserving Convexity Lemma. Let C i R n be a convex set for any i I where I is an index set (possibly infinite). Then the set i I C i is convex. Proof. In class Example: Consider the set P = {x R n : Ax b} where A R m n and b R m is convex. P is called a convex polyhedron and it is indeed convex. Why? Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 6 / 32

7 Algebraic Operations Preserving Convexity preservation under addition, cartesian product, forward and inverse linear mappings Theorem. 1. Let C 1, C 2,..., C k R n be convex sets and let µ 1, µ 2,..., µ k R. Then the set µ 1C 1 + µ 2C µ k C k is convex. 2. Let C i R k i, i = 1,..., m be convex sets. Then the cartesian product C 1 C 2 C m = {(x 1, x 2,..., x m) : x i C i, i = 1, 2,..., m} is convex. 3. Let M R n be a convex set and let A R m n. Then the set is convex. A(M) = {Ax : x M} 4. Let D R m be convex and let A R m n. Then the set is convex. A 1 (M) = {x R n : Ax D} Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 7 / 32

8 Convex Combinations Given m points x 1, x 2,..., x m R n, a convex combination of these m points is a vector of the form λ 1 x 1 + λ 2 x λ m x m, where λ 1, λ 2,..., λ m are nonnegative numbers satisfying λ 1 + λ λ m = 1. A convex set is defined by the property that any convex combination of two points from the set is also in the set. We will now show that a convex combination of any number of points from a convex set is in the set. Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 8 / 32

9 Convex Combinations Theorem.Let C R n be a convex set and let x 1, x 2,..., x m C. Then for any λ m, the relation m λ ix i C holds. Proof by induction on m. For m = 1 the result is obvious. The induction hypothesis is that for any m vectors x 1, x 2,..., x m C and any λ m, the vector m λ ix i belongs to C. We will now prove the theorem for m + 1 vectors. Suppose that x 1, x 2,..., x m+1 C and that λ m+1. We will show that z m+1 λ ix i C. If λ m+1 = 1, then z = x m+1 C and the result obviously follows. If λ m+1 < 1 then z = m m λ λ i ix i + λ m+1 x m+1 = (1 λ m+1 ) x i +λ m+1 x m+1. 1 λ m+1 }{{} v C and hence z = (1 λ m+1 )v + λ m+1 x m+1 C. v Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 9 / 32

10 The Convex Hull Definition. Let S R n. The convex hull of S, denoted by conv(s), is the set comprising all the convex combinations of vectors from S: { k } conv(s) λ i x i, : x 1, x 2,..., x k S, λ k. C conv(c) Figure: A nonconvex set and its convex hull Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 10 / 32

11 The Convex Hull The convex hull conv(s) is smallest convex set containing S. Proof. Lemma. Let S R n. If S T for some convex set T, then conv(s) T. Suppose that indeed S T for some convex set T. To prove that conv(s) T, take z conv(s). There exist x 1, x 2,..., x k S T (where k is a positive integer), and λ k such that z = k λ ix i. Since x 1, x 2,..., x k T, it follows that z T, showing the desired result. Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 11 / 32

12 We can assume that λ i > 0 for all i = 1, 2,..., k. If k n + 1, the result is proven. Otherwise, if k n + 2, then the vectors x 2 x 1, x 3 x 1,..., x k x 1, being more than n vectors in R n, are necessarily linearly dependent µ 2, µ 3,..., µ k not all zeros s.t. k i=2 µ i(x i x 1 ) = 0. Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 12 / 32 Carathéodory theorem Theorem. Let S R n and let x conv(s). Then there exist x 1, x 2,..., x n+1 S such that x conv ({x 1, x 2,..., x n+1 }), that is, there exist λ n+1 such that n+1 x = λ i x i. Proof. Let x conv(s). Then x 1, x 2,..., x k S and λ k s.t. x = k λ i x i.

13 Proof of Carathéodory Theorem Contd. Defining µ 1 = k i=2 µ i, we obtain that k µ i x i = 0, Not all of the coefficients µ 1, µ 2,..., µ k are zeros and k µ i = 0. There exists an index i for which µ i < 0. Let α R +. Then x = k λ i x i = k λ i x i + α k µ i x i = k (λ i + αµ i )x i. (1) We have k (λ i + αµ i ) = 1, so (1) is a convex combination representation iff λ i + αµ i 0 for all i = 1,..., k. (2). Since λ i > 0 for { all i, it follows that (2) is satisfied for all α [0, ε] where ε = min i:µi <0 λ i µ i }. Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 13 / 32

14 Proof of Carathéodory Theorem Contd. If we substitute { α = ε, then (2) still holds, but λ j + εµ j = 0 for j argmin µ } i. i:µ i <0 λ i This means that we found a representation of x as a convex combination of k 1 (or less) vectors. This process can be carried on until a representation of x as a convex combination of no more than n + 1 vectors is derived. Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 14 / 32

15 Example For n = 2, consider the four vectors ( 1 x 1 =, x 1) 2 = ( 1 2), x 3 = ( ( 2 2, x 1) 4 =, 2) and let x conv({x 1, x 2, x 3, x 4 }) be given by x = 1 8 x x x ( 13 ) 8 x 4 = Find a representation of x as a convex combination of no more than 3 vectors. In class Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 15 / 32

16 Convex Cones A set S is called a cone if it satisfies the following property: for any x S and λ 0, the inclusion λx S is satisfied. The following lemma shows that there is a very simple and elegant characterization of convex cones. Lemma. A set S is a convex cone if and only if the following properties hold: A. x, y S x + y S. B. x S, λ 0 λx S. Simple exercise Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 16 / 32

17 Examples of Convex Cones The convex polytope C = {x R n : Ax 0}, where A R m n. Lorentz Cone The Lorenz cone, or ice cream cone is given by {( ) } L n x = R n+1 : x t, x R n, t R. t nonnegative polynomials. set consisting of all possible coefficients of polynomials of degree n 1 which are nonnegative over R: K n = {x R n : x 1 t n 1 + x 2 t n x n 1 t + x n 0 t R} Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 17 / 32

18 The Conic Hull Definition. Given m points x 1, x 2,..., x m R n, a conic combination of these m points is a vector of the form λ 1 x 1 + λ 2 x λ m x n, where λ R m +. The definition of the conic hull is now quite natural. Definition. Let S R n. Then the conic hull of S, denoted by cone(s) is the set comprising all the conic combinations of vectors from S: { k } conv(s) λ i x i : x 1, x 2,..., x k S, λ R k +. Similarly to the convex hull, the conic hull of a set S is the smallest cone containing S. Lemma. Let S R n. If S T for some convex cone T, then cone(s) T. Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 18 / 32

19 Representation Theorem for Conic Hulls a similar result to Carathéodory theorem Conic Representation Theorem. Let S R n and let x cone(s). Then there exist k linearly independent vector x 1, x 2,..., x k S such that x cone ({x 1, x 2,..., x k }), that is, there exist λ R k + such that In particular, k n. x = k λ i x i. Proof very similar to the proof of Carathéodory theorem. See page 107 of the book for the proof. Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 19 / 32

20 Basic Feasible Solutions Consider the convex polyhedron. P = {x R n : Ax = b, x 0}, (A R m n, b R m ) the rows of A are assumed to be linearly independent. The above is a standard formulation of the constraints of a linear programming problem. Definition. x is a basic feasible solution (abbreviated bfs) of P if the columns of A corresponding to the indices of the positive values of x are linearly independent. Example.Consider the linear system: x 1 + x 2 + x 3 = 6 x 2 + x 4 = 3 x 1, x 2, x 3, x 4 0. Find all the basic feasible solutions. In class Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 20 / 32

21 Existence of bfs s Theorem.Let P = {x R n : Ax = b, x 0}, where A R m n and b R m. If P, then it contains at least one bfs. Proof. P b cone({a 1, a 2,..., a n }) where a i denotes the i-th column of A. By the conic representation theorem, there exist indices i 1 < i 2 <... < i k and k numbers y i1, y i2,..., y ik 0 such that b = k j=1 y i j a ij and a i1, a i2,..., a ik are linearly independent. Denote x = k j=1 y i j e ij. Then obviously x 0 and in addition A x = k y ij Ae ij = j=1 k y ij a ij = b. Therefore, x is contained in P and the columns of A corresponding to the indices of the positive components of x are linearly independent, meaning that P contains a bfs. j=1 Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 21 / 32

22 Topological Properties of Convex Sets Proof. Theorem.Let C R n be a convex set. Then cl(c) is a convex set. Let x, y cl(c) and let λ [0, 1]. There exist sequences {x k } k 0 C and {y k } k 0 C for which x k x and y k y as k. (*) λx k + (1 λ)y k C for any k 0. (**) λx k + (1 λ)y k λx + (1 λ)y. (*)+(**) λx + (1 λ)y cl(c). Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 22 / 32

23 The Line Segment Principle Theorem. Let C be a convex set and assume that int(c). Suppose that x int(c) and y cl(c). Then (1 λ)x + λy int(c) for any λ [0, 1). Proof. There exists ε > 0 such that B(x, ε) C. Let z = (1 λ)x + λy. We will show that B(z, (1 λ)ε) C. Let w B(z, (1 λ)ε). Since y cl(c), w 1 C s.t. w 1 y < (1 λ)ε w z. (3) λ Set w 2 = 1 1 λ (w λw 1). Then w 2 x = w λw1 1 λ x = 1 (w z) + λ(y w1) 1 λ 1 (3) ( w z + λ w1 y ) < ε, 1 λ Hence, since B(x, ε) C, it follows that w 2 C. Finally, since w = λw 1 + (1 λ)w 2 with w 1, w 2 C, we have that w C. Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 23 / 32

24 Convexity of the Interior Theorem. Let C R n be a convex set. Then int(c) is convex. Proof. If int(c) =, then the theorem is obviously true. Otherwise, let x 1, x 2 int(c), and let λ (0, 1). By the LSP, λx 1 + (1 λ)x 2 int(c), establishing the convexity of int(c). Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 24 / 32

25 Combination of Closure and Interior Lemma. Let C be a convex set with a nonempty interior. Then 1. cl(int(c)) = cl(c). 2. int(cl(c)) = int(c). Proof of 1. Obviously, cl(int(c)) cl(c) holds. To prove that opposite, let x cl(c), y int(c). Then x k = 1 k y + ( 1 1 k ) x int(c) for any k 1. Since x is the limit (as k ) of the sequence {x k } k 1 int(c), it follows that x cl(int(c)). For the proof of 2, see pages 109,110 of the book for the proof of Lemma 6.30(b). Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 25 / 32

26 Compactness of the Convex Hull of Convex Sets Proof. Theorem. Let S R n be a compact set. Then conv(s) is compact. M > 0 such that x M for any x S. Let y conv(s). Then there exist x 1, x 2,..., x n+1 S and λ n+1 for which y = n+1 λ ix i and therefore n+1 n+1 n+1 y = λ i x i λ i x i M λ i = M, establishing the boundedness of conv(s). To prove the closedness of conv(s), let {y k } k 1 conv(s) be a sequence converging to y R n. There exist x k 1, xk 2,..., xk n+1 S and λk n+1 such that n+1 y k = λ k i x k i. (4) Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 26 / 32

27 Proof Contd. By the compactness of S and n+1, it follows that {(λ k, x k 1, xk 2,..., xk n+1 )} k 1 has a convergent subsequence {(λ k j, x k j 1, xk j 2,..., xk j n+1 )} j 1 whose limit will be denoted by with λ n+1, x 1, x 2,..., x n+1 S Taking the limit j in (λ, x 1, x 2,..., x n+1 ) n+1 y kj = λ k j i x k j i, we obtain that y = n+1 λ ix i conv(s) as required. Example: S = {(0, 0) T } {(x, y) T : xy 1} Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 27 / 32

28 Closedness of the Conic Hull of a Finite Set Proof. Theorem. Let a 1, a 2,..., a k R n. Then cone({a 1, a 2,..., a k }) is closed. By the conic representation theorem, each element of cone({a 1, a 2,..., a k }) can be represented as a conic combination of a linearly independent subset of {a 1, a 2,..., a k }. Therefore, if S 1, S 2,..., S N are all the subsets of {a 1, a 2,..., a k } comprising linearly independent vectors, then cone({a 1, a 2,..., a k }) = N cone(s i ). It is enough to show that cone(s i ) is closed for any i {1, 2,..., N}. Indeed, let i {1, 2,..., N}. Then S i = {b 1, b 2,..., b m }, where b 1, b 2,..., b m are linearly independent. cone(s i ) = {By : y R m +}, where B is the matrix whose columns are b 1, b 2,..., b m. Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 28 / 32

29 Proof Contd. Suppose that x k cone(s i ) for all k 1 and that x k x. y k R m + such that x k = By k. (5) y k = (B T B) 1 B T x k. Taking the limit as k in the last equation, we obtain that y k ȳ where ȳ = (B T B) 1 B T x. ȳ R m +. Thus, taking the limit in (5), we conclude that x = Bȳ with ȳ R m +, and hence x cone(s i ). Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 29 / 32

30 Extreme Points Definition. Let S R n be a convex set. A point x S is called an extreme point of S if there do not exist x 1, x 2 S(x 1 x 2 ) and λ (0, 1), such that x = λx 1 + (1 λ)x 2. The set of extreme point is denoted by ext(s). For example, the set of extreme points of a convex polytope consists of all its vertices. 4 3 x 2 x 3 2 x x Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 30 / 32

31 Equivalence Between bfs s and Extreme Points Theorem. Let P = {x R n : Ax = b, x 0}, where A R m n has linearly independent rows and b R m. The x is a basic feasible solution of P if and only if it is an extreme point of P. Theorem 6.34 in the book. Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 31 / 32

32 Krein-Milman Theorem Theorem. Let S R n be a compact convex set. Then S = conv(ext(s)). Amir Beck Introduction to Nonlinear Optimization Lecture Slides - Convex Sets 32 / 32