IE 521 Convex Optimization
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1 Lecture 1: 16th January 2019
2 Outline 1 / 20
3 Which set is different from others? Figure: Four sets 2 / 20
4 Which set is different from others? Figure: Four sets 3 / 20
5 Interior, Closure, Boundary Definition. Let X be a nonempty set in R n. A point x 0 is called an interior point if r > 0, such that B(x 0, r) := {x : x x 0 2 r} X. A point x 0 is called a limit point if {x n } X, such that x n x 0 as n. Definition. Interior: int(x ) =the set of all interior point of X. Closure: cl(x ) =the set of all limit points of X. Boundary: (X ) = cl(x )/int(x ) Q. Let X = irrationals on [0, 1]. What are int(x ) and cl(x )? 4 / 20
6 Open and Closed Sets Definition. X is closed if cl(x ) = X ; X is open if int(x ) = X. Fact. int(x ) X cl(x ); X is closed iff X c = R n /X is open; α A X α is closed if X α is closed for all α A. n i=1 X i is closed if X i is closed for i = 1,..., n. Q. If X 1, X 2 are closed, is X 1 + X 2 closed? 5 / 20
7 Convex Set Definition A set X R n is convex if x, y X, λ [0, 1] λx + (1 λ)y X. In another word, the line segment [x, y] that connects any two elements x, y lies entirely in the set. (a) convex (b) non-convex Figure: of convex and non-convex sets 6 / 20
8 Convex, Conic, Affine, and Linear Combinations Definition. Given any elements x 1,..., x k, the combination λ 1 x λ k x k is called Convex: if λ i 0, i = 1,..., k and λ λ k = 1; Conic: if λ i 0, i = 1,..., k; Affine: if λ λ k = 1; Linear: if λ i R, i = 1,..., k. 7 / 20
9 , Cones, Affine and Linear Subspaces Definition. A set is convex if all convex combinations of its elements are in the set; A set is a convex cone if all conic combinations of its elements are in the set; A set is a affine subspace if all affine combinations of its elements are in the set; A set is a linear subspace if all linear combinations of its elements are in the set. Clearly, a linear subspace is always a convex cone; a convex cone is always a convex set. Note: Cones vs. Convex cones. 8 / 20
10 Convex, Conic, Affine Hulls Definition. Given any set X, we define Convex hull of X : Conv(X ) = set of all convex combinations of points in X. Conic hull of X : Cone(X ) = set of all conic combinations of points in X. Affine hull of X : Aff(X ) = set of all affine combinations of points in X. Figure: of convex hulls 9 / 20
11 Properties of Proposition. 1. A convex hull is always convex. 2. If X is convex, then Conv(X ) = X. 3. For any set X, Conv(X ) is the smallest convex set that contains X. 10 / 20
12 of Example 1. Simple sets: Hyperplane: {x R n : a T x = b} Halfspace: {x R n : a T x b} Affine space: {x R n : Ax = b} Polyhedron: {x R n : Ax b} Simplex: {x R n : x 0, n i=1 x i = 1}. Example 2. Euclidean balls: {x R n : x a 2 r}. Example 3. Ellipsoid: {x R n : (x a) T Q(x a) r 2 } where Q 0 and is symmetric. 11 / 20
13 of Convex Cones Example 1. Example 2. Positive Orthant: {x R n : x 0} Norm cones: {(x, t) R n+1 : x 2 t} Example 3. Positive semidefinite matrices: S n + := {X S n : X 0} 12 / 20
14 Operations that Preserves Convexity Intersection If X α, α A are convex sets, then so is α A X α. Cartesian product: If X i R n i, i = 1,..., k are convex, then so is X 1 X k. Weighted summation: If X i R n, i = 1,..., k convex, then so is α 1 X α k X k. 13 / 20
15 Operations that Preserves Convexity Affine image: If X R n is a convex set and A(x) : x Ax + b is an affine mapping from R n to R k, then so is A(X ) := {Ax + b : x X }. Proof: Let y 1, y 2 A(X ) x 1, x 2 X such that y 1 = Ax 1 + b and y 2 = Ax 2 + b. For λ [0, 1], λy 1 + (1 λ)y 2 = A(λx 1 + (1 λ)x 2 ) + b A(X ) because λx 1 + (1 λ)x 2 X. 14 / 20
16 Operations that Preserves Convexity Inverse affine image: If X R n is a convex set and A(y) : y Ay + b is an affine mapping from R k to R n, then so is Proof: self-exercise. A 1 (X ) := {y : Ay + b X }. Example. The solution set of linear matrix inequality: {x x 1 A x k A k B} where A i, B are positive semidefinite matrices. 15 / 20
17 Nice Properties of Proposition. Let X be convex with nonempty interior. Then If x 0 int(x ) and x cl(x ), then [x 0, x) int(x ). Moreover, int(x ) is dense in cl(x ). Remark. In general, int(x ) and cl(x ) can differ dramatically. If X = irrationals on [0, 1], int(x ) =, cl(x ) = [0, 1]. Q. What happens if X is convex but int(x ) =? 16 / 20
18 Nice Properties of Definition. (Relative Interior and Dimension) rint(x ) = {x : r > 0, s.t. B(x, r) Aff(X ) X } dim(x ) = dim(aff(x )) Fact. For a convex and nonempty set, rint(x ). Proposition. Let X be a nonempty convex set. Then a) int(x ),cl(x ), rint(x ) are convex b) x 0 rint(x ), x cl(x ) [x 0, x) rint(x ), λ (0, 1] c) cl(rint(x )) = cl(x ) d) rint(cl(x )) = rint(x ) 17 / 20
19 Question Suppose there are 100 different kinds of herbal tea, everyone of them is a blend of 25 herbs. Donald wants a particular mixture of all herbal teas with equal proportions. What s the least number of teas he should buy? 18 / 20
20 Carathéodory. Let X R n be non empty and dim(x ) = d n. Every point x Conv(X ) is a convex combination of at most (d + 1) points, i.e. Conv(X ) = { d+1 d+1 λ i x i : x i X, λ i 0, λ i = 1 i=1 i=1 Proof: Suppose the minimal representation of x Conv(X ) has m d + 1 terms, x = m i=1 α ix i, where α i 0, m i=1 α i = 1.The system of linear equations { m }. i=1 δ ix i = 0 m i=1 δ i = 0 has non trivial solution. Rewrite x = m i=1 (α i tδ i )x i. Let λ i (t) = (α i tδ i ), i, we have { } λi (t) = 1. Let t = min αi δ i, δ i > 0 := α j δ j, then λ i (t ) > 0, i j and λ j (t ) = 0. This leads to a smaller representation of x. 19 / 20
21 References Boyd & Vandenberghe, Chapter Ben-Tal & Nemirovski, Chapter / 20
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