Lecture 2. Topology of Sets in R n. August 27, 2008
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1 Lecture 2 Topology of Sets in R n August 27, 2008
2 Outline Vectors, Matrices, Norms, Convergence Open and Closed Sets Special Sets: Subspace, Affine Set, Cone, Convex Set Special Convex Sets: Hyperplane, Polyhedral Set, Ball Special Cones: Normal Cone, Dual Cone, Tangent Cone Some Operations with Sets: Set Closure, Interior, Boundary Convex Hull, Conical Hull, Affine Hull Relative Closure, Interior, Boundary Functions Special Functions: Linear, Quadratic Game Theory: Models, Algorithms and Applications 1
3 Notation: Vectors and Matrices We consider the space of n-dimensional real vectors denoted by R n Vector x R n is viewed as a column: x = The prime denotes a transpose: x = [x 1,..., x n ] For a matrix A, we use a ij or [A] ij to denote its (i, j)-th entry The matrix A is positive semidefinite (A 0) when x 1. x n x Ax 0 for all x R n The matrix A is positive definite (A > 0) when x Ax > 0 for all x R n, x 0 Game Theory: Models, Algorithms and Applications 2
4 Inner Product and Norms For vectors x R n and y R n, the inner product is Norm (length) of a vector: x y = n i=1 x i y i Euclidean norm x = x x = ni=1 x 2 i (l 2 -norm) Sum-norm x 1 = n i=1 x i (l 1 -norm) Max-norm x = max 1 i n x i (l -norm) Matrix norm induced by (Euclidean) vector norm A = sup x 0 = sup x =1 Ax x Ax x Game Theory: Models, Algorithms and Applications 3
5 Norm Properties, Distance, Orthogonal Vectors Properties of a norm Nonnegativity: x 0 for every x R n Uniqueness of the zero-value: x = 0 if and only if x = 0 Homogeneity: Triangle relation: λx = λ x for any scalar λ and any x R n x + y x + y for every x and y R n (Euclidean) Distance between two vectors x R n and y R n x y = n i=1 (x i y i ) 2 Vectors x and y are orthogonal when x y = 0 For orthogonal vectors x and y: Pythagorean Theorem x + y 2 = x 2 + y 2 For any vectors x and y: Schwartz Inequality: x y x y Game Theory: Models, Algorithms and Applications 4
6 Vector Sequence Given a vector sequence {x k } R n, we say that The sequence is bounded when for some scalar C, we have x k C for all k The sequence converges to a vector x R n when lim k x k x = 0 (x k x) Vector y is an accumulation point of the sequence when there is a subsequence {x ki } such that x ki y as i Example: x k = (1 + ( 1) k a) k with 0 < a < 1. Accum. points? Bolzano Theorem A bounded sequence {x k } R n has at least one limit point Game Theory: Models, Algorithms and Applications 5
7 Set Topology Typically denoted by capital letters C, X, Y, Z, etc. Let X be a set in R n. A vector x is an accumulation (or a limit) point of the set X when there is a sequence {x k } X such that x k x The set X is closed when it contains all of its accumulation points The set X is open when its complement set X c = {x R n x X} is closed R n and are the only sets that are both open and closed The set X is bounded when for some C: x C for all x X (Th.) The set X is compact when it is both closed and bounded Let {X i i I} be a family of closed sets X i R n The intersection i I X i is closed When the set I is finite (I = {1,..., m}) the union i I X i is closed Game Theory: Models, Algorithms and Applications 6
8 Special Sets Let X be a set in R n. We say that X is: subspace when αx + βy X for every x, y X and any α, β R Example: {x x e = 0} where e R n with e i = 1 for all i Is a subspace a closed set? Bounded? affine when it is a translated subspace: X = x + S for an x X and a subspace S cone when λx X for every λ 0 and every x X convex when αx + (1 α)y X for any x, y X and α [0, 1] Game Theory: Models, Algorithms and Applications 7
9 Convex Sets Is a subspace convex set? Affine set? Special convex sets: (Euclidean) Ball centered at x 0 R n with a radius r > 0 {x R n x x 0 r} (closed ball) {x R n x x 0 < r} (open ball) A closed ball is convex; An open ball is convex Game Theory: Models, Algorithms and Applications 8
10 More Convex Sets Half-space with normal vector a 0: {x a x b} Hyperplane with normal vector a 0: {x a x = b} (b is a scalar) Polyhedral set (given by finitely many linear inequalities): {x a i x b i, i = 1,..., m} = {x Ax b} A is an m n matrix and b R m Game Theory: Models, Algorithms and Applications 9
11 Dual Cone Let K R n be a cone. Def. A dual cone of K is the following set K = {d R n d y 0 for all y K} Dual cone appears in formulations of complementarity problems Dual cone is always closed and convex (regardless of K) Game Theory: Models, Algorithms and Applications 10
12 Normal Cone of a Set Let X R n be a nonempty set, and let ˆx X. Def. A normal cone of the set X at the point ˆx is the following set N(ˆx; X) = {y R n y (x ˆx) 0 for all x X} Vectors in this set are called normal vectors to the set X at ˆx ~ ~ x X N(x ; X ) = {0} x^ ^ N(x ; X ) Normal cone plays an important role in optimality conditions Normal cone is always closed and convex (regardless of X) Game Theory: Models, Algorithms and Applications 11
13 Tangent Cone of a Set Let X R n be a nonempty set, and let ˆx X. Def. A tangent cone of the set X at the point ˆx is the following set T (ˆx; X) = { y R n y = lim k x k ˆx λ k with {x k } X, {λ k } R ++ s. t. x k ˆx, λ k 0} Vectors in this set are called tangent vectors to the set X at ˆx Prop. The tangent cone T (ˆx; X) is equivalently given by T (ˆx; X) = { βy R n y = lim k x k ˆx x k ˆx with {x k} X, x k ˆx, x k ˆx, β 0} Game Theory: Models, Algorithms and Applications 12
14 Tangent Cone Illustration ^ T(x ; X ) ~ x X x^ Tangent cone T (ˆx; X) for any set X and any ˆx X: plays an important role in optimality conditions always closed regardless of the properties of X when X is convex, then T (ˆx; X) is convex Game Theory: Models, Algorithms and Applications 13
15 Tangent and Normal Cone Let X R n and let ˆx X. Then, we have N(ˆx; X) T (ˆx; X) ^ T(x ; X )* ~ x X ^ N(x ; X ) x^ Prop. If X is convex, then T (ˆx; X) = N(ˆx; X) for all ˆx X Game Theory: Models, Algorithms and Applications 14
16 Set Interior, Closure and Boundary Let X R n. A vector ˆx X is an interior point of X if there exist a ball B(ˆx, r) such that B(ˆx, r) X The interior of the set X is the set of all interior points of X; it is denoted by int(x) A vector ˆx X is a accumulation point of X if there exist a sequence {x k } X converging to ˆx The closure of the set X is the set of all accumulation points of X; it is denoted by cl(x) The boundary of the set X is the intersection of the closure set of X and the closure set of its complement (R n \ X), i.e., bd(x) = cl(x) cl(r n \ X) Example: X = {(x 1, x 2 ) R 2 0 x 1 1, 0 < x 2 < 1} Game Theory: Models, Algorithms and Applications 15
17 Convex and Conical Hulls A convex combination of vectors x 1,..., x m is a vector of the form α 1 x α m x m α i 0 for all i and m i=1 α i = 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 x 1,..., x m is a vector of the form λ 1 x λ m x m with λ i 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) Game Theory: Models, Algorithms and Applications 16
18 Affine Hull An affine combination of vectors x 1,..., x m is a vector of the form t 1 x t m x m with m i=1 t i = 1, t i 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)) Game Theory: Models, Algorithms and Applications 17
19 Relative Interior, Closure, and Boundary Let X R n. A vector ˆx X is a relative interior point of X if there exist a ball B(ˆx, r) such that B(ˆx, r) aff(x) X The relative interior of the set X is the set of all interior points of X; it is denoted by rint(x) The relative boundary of the set X is the intersection of the affine hull aff(x) and the boundary bd(x) Example: X = {(x 1, x 2 ) R 2 0 x 1 1, x 2 = 1} aff(x) =? Game Theory: Models, Algorithms and Applications 18
20 Mappings Let X R n. We write F : X R m to denote a mapping F from the set X to R m F is a function when F : X R (m = 1) We refer to X as a domain of the mapping F The image of X under mapping F is the following set F (X) = {y R m y = F (x) for some x X} The inverse image of Y R m under mapping F is the following set F 1 (Y ) = {x X F (x) Y } Game Theory: Models, Algorithms and Applications 19
21 Special Mappings and Functions A mapping F : X R m with X R n is affine when for all x X, F (x) = Ax + b for some matrix A and a vector b R m Example: Space translation (by a given x 0 R n ) is an affine mapping from R n to R n : F (x) = x + x 0 for all x R n linear when for all x X, we have F (x) = Ax for a matrix A Example: For a coordinate subspace S = {x R n x i = 0, for i I}, where I is a subset of coordinate indices (I {1,..., n}, the projection on S is a linear mapping from R n to R n : F (x) = P x with [P ] ii = 1 for i I and P ij = 0 otherwise Game Theory: Models, Algorithms and Applications 20
22 Special Functions A function f : R n R is quadratic when for all x f(x) = x Qx + a x + b for a matrix Q, a R n and b R affine when for all x f(x) = a x + b for a vector a R n and b R linear when for all x f(x) = a x for a vector a R n Game Theory: Models, Algorithms and Applications 21
23 The material for this lecture: References for this lecture (B) Bertsekas D.P. Nonlinear Programming See Appendix there for topology in R n and linear algebra (BNO) Bertsekas, Nedić, Ozdaglar Convex Analysis and Optimization Chapter 1 for convex sets and functions see also lecture slides for Convex Optimization (Lectures 2 5) at (FP) Facchinei and Pang Finite Dimensional..., Vol I Chapter 1 for Normal Cone, Dual Cone, and Tangent Cone NOTE!!! Normal cone definition used in (FP) is different than that of (B,BNO) Game Theory: Models, Algorithms and Applications 22
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