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Lecture 2 — Convexity TMA947 / MMG621 – Nonlinear optimisation Lecture 2 TMA947 / MMG621 — Nonlinear optimisation Lecture 2 — Convexity Emil Gustavsson October 28, 2013 Convex sets We begin by defining the notion of a convex set. Definition (convex set). The set S ⊆ Rn is convex if x1, x2 ∈ S ⇒ λx1 + (1 − λ)x2 ∈ S. λ ∈ (0, 1) A set is thus convex if all convex combination of any two points in the set lies in the set, see Figure 1. x2 λx1 + (1 − λ)x2 S x1 Figure 1: A convex set Examples: – The empty set is a convex set. – The set {x ∈ Rn |kxk≤ a} is convex for any a ∈ R. – The set {x ∈ Rn |kxk = a} is non-convex for any a> 0. – The set {0, 1, 2, 3} is non-convex. Rn Proposition. Let Sk ⊆ , k ∈ K be a collection of convex sets. Then the intersection k∈K Sk is also a convex set. T Proof. See Proposition 3.3. in the book. 1 TMA947 / MMG621 – Nonlinear optimisation Lecture 2 We define the affine hull of a finite set V = {v1, v2,..., vk} as k 1 k aff V := λ1v + ··· + λkv λ1,...,λk ∈ R, λi =1 ( i ) =1 X We define the convex hull of a finite set V = {v1, v2,..., vk} as k 1 k conv V := λ1v + ··· + λkv λ1,...,λk ≥ 0, λi =1 ( i ) =1 X The sets are defined by all possible affine (convex) combinations of the k points. v2 v2 v2 v1 v1 v1 V aff V conv V 1 2 Figure 2: The set V = {v , v }, the affine hull of V , and the convex hull of V . In general, we can define the convex hull of a set S as a) the unique minimal convex set containing S, b) the intersection of all convex sets containing S, or c) the set of all convex combinations of points in S. Any point x ∈ conv S, where S ⊆ Rn can thus be expressed as a convex combination of points in S. But how many points are required? The answer is, according to the following theorem, n +1 points. Theorem (Caratheodory’s theorem). Let x ∈ conv S, where S ⊆ Rn. Then x can be expressed as a convex combination of n +1 or fewer points of S. Proof. See Theorem 3.11. in the book. We now define the notion of a polytope. Definition (polytope). A set P is a polytope if it is the convex hull of finitely many points in Rn. A cube and a tetrahedron are examples of polytopes in R3. In Figure 3, a polytope in R2 generated by seven points is illustrated. 2 TMA947 / MMG621 – Nonlinear optimisation Lecture 2 v2 v6 v5 v3 v1 v7 v4 Figure 3: A polytope generated by seven point. The red dots are the extreme points Definition (extreme point). A point v of a convex set P is an extreme point if whenever v = λx1 + (1 − λ)x2 x1, x2 ∈ P =⇒ v = x1 = x2. λ ∈ (0, 1) An extreme point of P is thus a point in P that can not be represented as a convex combination of two other points, see Figure 3 for an example. We can now formulate the following intuitive theorem. Theorem. Let P be the polytope conv V , where V = {v1,..., vk}⊂ Rn. Then P is equal to the convex hull of its extreme points. The theorem basically says that the interesting points in a polytope are the extreme points. By combining the two last theorems, we can deduce that any point in a polytope P can be described as a convex combination of at most n +1 extreme points of P . This is often denoted as the inner representation of P . We now try to describe a polytope as linear inequalities instead of extreme points (exterior representation). Definition (polyhedron). A set P is a polyhedron if there exists a matrix A ∈ Rm×n and a vector b ∈ Rm such that P = {x ∈ Rn | Ax ≤ b}. – Ax ≤ b ⇐⇒ aix ≤ bi, i =1,...,m. (ai row i of A) n – {x ∈ R | aix ≤ bi}, i =1,...,m are half-spaces, so – P is the intersection of m half-spaces. 1 2 6 – Figure 4 shows the bounded polyhedron defined by A = −2 1 and b = −2 0 −1 −1 −2 1 −2 – Figure 5 shows the undounded polyhedron defined by A = and b = . 0 −1 −1 To make the distinction between a polytope and a polyhedron clearer, we have that 3 TMA947 / MMG621 – Nonlinear optimisation Lecture 2 x1 2x1 − x2 =2 x1 +2x2 =6 3 2 P 1 x2 =1 x2 1 2 3 4 5 Figure 4: A bounded polyhedron described by three linear inequalities. x1 2x1 − x2 =2 3 2 P 1 x2 =1 x2 1 2 3 4 5 Figure 5: An unbounded polyhedron described by two linear inequalities. – A polytope = The convex hull of finitely many points. – A polyhedron = The intersection of finitely many half-spaces. We can now define the extreme points of a polyhedron algebraically Theorem. Let x¯ ∈ P = {x ∈ Rn | Ax ≤ b}, where rank A = n and b ∈ Rm. Further, let A¯ x¯ = b¯ be the equality subsystem of Ax ≤ b. Then x¯ is an extreme point of P if and only if rank A¯ = n Proof. See Theorem 3.21 in the book. Note that the equality subsystem is obtained if one strikes out all rows i with aix¯ <bi, and require equality for the rest of the rows. That the equality subsystem has rank n basically means that there should be at least n linearly independent half-spaces going through the point in order for it to be an extreme point. In Figure 6, two half-spaces goes through the point x¯. Hence, it is an extreme point. In Figure 7, however, only one half-space goes through x¯, meaning that it is not an extreme point. How many extreme points can there exist in a polyhedron P = {x ∈ Rn | Ax ≤ b}? How many m ways can we choose n linearly independent rows from the m existing rows? At most . n Definition (cone). A set C ⊆ Rn is a cone if λx ∈ C whenever x ∈ C and λ> 0. 4 TMA947 / MMG621 – Nonlinear optimisation Lecture 2 x1 2x1 − x2 =2 x1 +2x2 =6 3 2 P 1 x2 =1 x¯ = (3/2, 1)T x2 1 2 3 4 5 Figure 6: x¯ = (3/2, 1)T is an extreme point of P x1 2x1 − x2 =2 x1 +2x2 =6 3 2 P 1 x2 =1 x¯ = (2, 1)T x2 1 2 3 4 5 Figure 7: x¯ = (2, 1)T is not an extreme point of P Example: The set {x ∈ Rn | Ax ≤ 0} is a polyhedral cone. In Figure 11 one convex and one non-convex cone are illustrated. C C C (a) A convex cone (b) A non-convex cone Figure 8: Two cones. We can now formulate a very important theorem which basically says that a polyhedron = a polytope + a polyhedral cone Theorem (The representation theorem). Let the polyhedron Q = {x ∈ Rn | Ax ≤ b} and let {v1,..., vk} be its extreme points. Define P := conv {v1,..., vk} and C := {x ∈ Rn | Ax ≤ 0}. Then Q = P + C = {x ∈ Rn | x = u + v for some u ∈ P and v ∈ C} Proof. See Theorem 3.26 in the book. 5 TMA947 / MMG621 – Nonlinear optimisation Lecture 2 T x2 y = (2, 2) T π x = α ⇔ x1 + x2 =2 S x1 T T T Figure 9: The hyperplane π x = α separates the point y = (2, 2) from the unit disc. π = (1, 1) and α = 2. Now we will present a very useful theorem for convex sets which says that: "If a point y does not lie in a closed convex set S, then there exist a hyperplane separating y from S". Mathematically, this amounts to the following. Theorem (The separation theorem). Suppose that the set S ⊆ Rn is closed and convex, and that the n point y does not lie in S. Then there exist α ∈ R and π 6= 0 such that πT y > α and πT x ≤ α for all x ∈ S. Proof. Later in the course. One last important theorem related to convex set is Farkas’ Lemma. Theorem (Farkas’ Lemma). Let A ∈ Rm×n and b ∈ Rm. Then exactly one of the systems Ax = b, (A) x ≥ 0, and T A π ≤ 0, T (B) b π > 0, has a feasible solution, and the other is inconsistent. Proof. See Theorem 3.37 in the book. Farkas’ Lemma says that: (A) Either b lies in the cone spanned by the columns of A, i.e., b = Ax, for some x ≥ 0, (B) or, b does not lie in the cone, meaning that there exists a hyperplane π separating b from the cone, i.e., ATπ ≤ 0, bTπ > 0. 6 TMA947 / MMG621 – Nonlinear optimisation Lecture 2 y y = f(x) f(x1) λf(x1)+(1 − λ)f(x2) f(x2) f(λx1 + (1 − λ)x2) x x1 λx1 + (1 − λ)x2 x2 Figure 10: A convex function Convex functions Definition (convex function). Suppose S ⊆ Rn is convex. A function f : Rn → R is convex on S if x1, x2 ∈ S =⇒ f λx1 + (1 − λ)x2 ≤ λf(x1)+(1 − λ)f(x2). λ ∈ (0, 1) – A function is strictly convex on S if < holds in place of ≤ for all x1 6= x2. – A function f is concave if −f is convex. The linear interpolation between two points on the function never is lower than the function. Two important examples: – f(x)= cTx + d, where c ∈ Rn, d ∈ R is both convex and concave.
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