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Lecture 2: Convex sets August 28, 2008 Lecture 2 Outline • Review basic topology in Rn • Open Set and Interior • Closed Set and Closure • Dual Cone • Convex set • Cones • Aﬃne sets • Half-Spaces, Hyperplanes, Polyhedra • Ellipsoids and Norm Cones • Convex, Conical, and Aﬃne Hulls • Simplex • Verifying Convexity Convex Optimization 1 Lecture 2 Topology Review Let {xk } be a sequence of vectors in Rn Def. The sequence {xk } ⊆ Rn converges to a vector x ∈ Rn when ˆ xk − x tends to 0 as k → ∞ ˆ • Notation: When {xk } converges to a vector x, we write xk → x ˆ ˆ • The sequence {xk } converges x ∈ Rn if and only if for each component ˆ i: the i-th components of xk converge to the i-th component of x ˆ |xi − xi| tends to 0 as k → ∞ k ˆ Convex Optimization 2 Lecture 2 Open Set and Interior Let X ⊆ Rn be a nonempty set Def. The set X is open if for every x ∈ X there is an open ball B(x, r) that entirely lies in the set X , i.e., for each x ∈ X there is r > 0 s.th. for all z with z − x < r, we have z ∈ X Def. A vector x0 is an interior point of the set X , if there is a ball B(x0, r) contained entirely in the set X Def. The interior of the set X is the set of all interior points of X , denoted by (X) • How is (X) related to X ? • Example X = {x ∈ R2 | x1 ≥ 0, x2 > 0} (X) = {x ∈ R2 | x1 > 0, x2 > 0} (S) of a probability simplex S = {x ∈ Rn | x 0, e x = 1} Th. For a convex set X , the interior (X) is also convex Convex Optimization 3 Lecture 2 Closed Set Def. The complement of a given set X ⊆ Rn is the set of all vectors that do not belong to X : the complement of X = {x ∈ Rn | x ∈ X} = Rn \ X / Def. The set X is closed if its complement Rn \ X is open • Examples: Rn and ∅ (both are open and closed) {x ∈ R2 | x1 ≥ 0, x2 > 0} is open or closed? hyperplane, half-space, simplex, polyhedral sets? • The intersection of any family of closed set is closed • The union of a ﬁnite family of closed set is closed • The sum of two closed sets is not necessarily closed • Example: C1 = {(x1, x2) | x1 = 0, x2 ∈ R} C2 = {(x1, x2) | x1x2 ≥ 1, x1 ≥ 0} C1 + C2 is not closed! • Fact: The sum of a compact set and a closed set is closed Convex Optimization 4 Lecture 2 Closure Let X ⊆ Rn be a nonempty set Def. A vector x is a closure point of a set X if there exists a sequence ˆ {xk } ⊆ X such that xk → x ˆ ˆ Closure points of X = {(−1)n/n | n = 1, 2, . . .}, X = {1 − x | x ∈ X}? • Notation: The set of closure points of X is denoted by cl(X) • What is relation between X and cl(X)? Th. A set is closed if and only if it contains its closure points, i.e., X is closed iﬀ cl(X) ⊂ X Th. For a convex set, the closure cl(X) is convex Convex Optimization 5 Lecture 2 Boundary Let X ⊆ Rn be a nonempty set Def. The boundary of the set X is the set of closure points for both the set X and its complement Rn \ X , i.e., bd(X) = cl(X) ∩ cl(Rn \ X) • Example X = {x ∈ Rn | g1(x) ≤ 0, . . . , gm(x) ≤ 0}. Is X closed? What constitutes the boundary of X ? Convex Optimization 6 Lecture 2 Dual Cone Let K be a nonempty cone in Rn Def. The dual cone of K is the set K ∗ deﬁned by K ∗ = {z | z x ≥ 0 for all x ∈ K} • The dual cone K ∗ is a closed convex cone even when K is neither closed nor convex • Let S be a subspace. Then, S ∗ = S ⊥. • Let C be a closed convex cone. Then, (C ∗)∗ = C . • For an arbitrary cone K , we have (K ∗)∗ = cl(conv(K)). Convex Optimization 7 Lecture 2 Convex set • A line segment deﬁned by vectors x and y is the set of points of the form αx + (1 − α)y for α ∈ [0, 1] • A set C ⊂ Rn is convex when, with any two vectors x and y that belong to the set C , the line segment connecting x and y also belongs to C Convex Optimization 8 Lecture 2 Examples Which of the following sets are convex? • The space Rn • A line through two given vectors x and y l(x, y) = {z | z = x + t(y − x), t ∈ R} • A ray deﬁned by a vector x {z | z = λx, λ ≥ 0} • The positive orthant {x ∈ Rn | x 0} ( componentwise inequality) • The set {x ∈ R2 | x1 > 0, x2 ≥ 0} • The set {x ∈ R2 | x1x2 = 0} Convex Optimization 9 Lecture 2 Cone A set C ⊂ Rn is a cone when, with every vector x ∈ C , the ray {λx | λ ≥ 0} belongs to the set C • A cone may or may not be convex • Examples: {x ∈ Rn | x 0} {x ∈ R2 | x1x2 ≥ 0} For a two sets C and S , the sum C + S is deﬁned by C + S = {z | z = x + y, x ∈ C, y ∈ S} (the order does nor matter) Convex Cone Lemma: A cone C is convex if and only if C + C ⊆ C Proof: Pick any x and y in C , and any α ∈ [0, 1]. Then, αx and (1 − α)y belong to C because... . Using C + C ⊆ C , it follows that ... Reverse: Let C be convex cone, and pick any x, y ∈ C . Consider 1/2(x + y)... Convex Optimization 10 Lecture 2 Aﬃne Set A set C ⊂ Rn is a aﬃne when, with every two distinct vectors x, y ∈ C , the line {x + t(y − x) | t ∈ R} belongs to the set C • An aﬃne set is always convex • A subspace is an aﬃne set A set C is aﬃne if and only if C is a translated subspace, i.e., C = S + x0 for some subspace S and some x0 ∈ C Dimension of an aﬃne set C is the dimension of the subspace S Convex Optimization 11 Lecture 2 Hyperplanes and Half-spaces Hyperplane is a set of the form {x | a x = b} for a nonzero vector a Half-space is a set of the form {x | a x ≤ b} with a nonzero vector a The vector a is referred to as the normal vector • A hyperplane in Rn divides the space into two half-spaces {x | a x ≤ b} and {x | a x ≥ b} • Half-spaces are convex • Hyperplanes are convex and aﬃne Convex Optimization 12 Lecture 2 Polyhedral Sets A polyhedral set is given by ﬁnitely many linear inequalities C = {x | Ax b} where A is an m × n matrix • Every polyhedral set is convex • Linear Problem minimize cx subject to Bx ≤ b, Dx = d The constraint set {x | Bx ≤ b, Dx = d} is polyhedral. Convex Optimization 13 Lecture 2 Ellipsoids Let A be a square (n × n) matrix. • A is positive semideﬁnite when x Ax ≥ 0 for all x ∈ Rn • A is positive deﬁnite when x Ax > 0 for all x ∈ Rn, x = 0 An ellipsoid is a set of the form E = {x | (x − x0) P −1(x − x0) ≤ 1} where P is symmetric and positive deﬁnite • x0 is the center of the ellipsoid E • A ball {x | x − x0 ≤ r} is a special case of the ellipsoid (P = r2I ) • Ellipsoids are convex Convex Optimization 14 Lecture 2 Norm Cones A norm cone is the set of the form C = {(x, t) ∈ Rn × R | x ≤ t} • The norm · can be any norm in Rn • The norm cone for Euclidean norm is also known as ice-cream cone • Any norm cone is convex Convex Optimization 15 Lecture 2 Convex and Conical Hulls A convex combination of vectors x1, . . . , xm is a vector of the form m α 1 x1 + . . . + α m xm αi ≥ 0 for all i and i=1 αi =1 The convex hull of a set X is the set of all convex combinations of the vectors in X , denoted conv(X) A conical combination of vectors x1, . . . , xm is a vector of the form λ1x1 + . . . + λmxm with λi ≥ 0 for all i The conical hull of a set X is the set of all conical combinations of the vectors in X , denoted by cone(X) Convex Optimization 16 Lecture 2 Aﬃne Hull An aﬃne combination of vectors x1, . . . , xm is a vector of the form m t1x1 + . . . + tmxm with i=1 ti = 1, ti ∈ R for all i The aﬃne hull of a set X is the set of all aﬃne combinations of the vectors in X , denoted aﬀ (X) The dimension of a set X is the dimension of the aﬃne hull of X dim(X) = dim(aﬀ (X)) Convex Optimization 17 Lecture 2 Simplex A simplex is a set given as a convex combination of a ﬁnite collection of vectors v0, v1, . . . , vm: C = conv{v0, v1 . . . , vm} The dimension of the simplex C is equal to the maximum number of linearly independent vectors among v1 − v0, . . . , vm − v0. Examples • Unit simplex {x ∈ Rn | x 0, e x ≤ 1}, e = (1, . . . , 1), dim -? • Probability simplex {x ∈ Rn | x 0, e x = 1}, dim -? Convex Optimization 18 Lecture 2 Practical Methods for Establishing Convexity of a Set Establish the convexity of a given set X • The set is one of the “recognizable” (simple) convex sets such as polyhedral, simplex, norm cone, etc • Prove the convexity by directly applying the deﬁnition For every x, y ∈ X and α ∈ (0, 1), show that αx + (1 − α)y is also in X • Show that the set is obtained from one of the simple convex sets through an operation that preserves convexity Convex Optimization 19 Lecture 2 Operations Preserving Convexity Let C ⊆ Rn, C1 ⊆ Rn, C2 ⊆ Rn, and K ⊆ Rm be convex sets. Then, the following sets are also convex: • The intersection C1 ∩ C2 = {x | x ∈ C1 and x ∈ C2} • The sum C1 + C2 of two convex sets • The translated set C + a • The scaled set tC = {tx | x ∈ C} for any t ∈ R • The Cartesian product C1 × C2 = {(x1, x2) | x1 ∈ C1, x2 ∈ C2} • The coordinate projection {x1 | (x1, x2) ∈ C for some x2} • The image AC under a linear transformation A : Rn → Rm: AC = {y ∈ Rm | y = Ax for some x ∈ C} • The inverse image A−1K under a linear transformation A : Rn → Rm: A−1K = {x ∈ Rn | Ax ∈ K} Convex Optimization 20

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