Topological properties for sets and functions

Chapter 2 Topological properties for sets and functions Chapter 2. Topological properties for sets and functions tvnguyen (University of Science) Convex Optimization 26 / 108 Chapter 2 Topological properties for sets and functions Relative Interior of a Convex Set The interior of a subset C of IR n is the union of all open sets (of IR n ) contained in C. Since any union of open sets is open, the interior is also the largest open set (of IR n ) contained in C. The interior of C is denoted by intC . From this definition, we have x ∈ int C ⇔ ∃δ > 0 such that B(x, δ) ⊆ C . Many nonempty convex sets have an empty interior. For example, in IR 2 , int [a, b] = ∅. Similarly, in IR 3 , the interior of a triangle ABC is empty. So our aim is to define a substitute to the interior of a convex set, called the relative interior, in such a way that the relative interior of any nonempty convex set is nonempty. To define the relative interior, we need the concept of affine set. tvnguyen (University of Science) Convex Optimization 27 / 108 Chapter 2 Topological properties for sets and functions Affine Sets. Definition Let A be a nonempty subset of IR n . A is affine if ∀x, y ∈ A, ∀α ∈ IR αx + (1 − α)y ∈ A. The vector αx + (1 − α)y is called an affine combination of x and y. The line passing through two points x and y is defined by {αx + (1 − α)y | α ∈ IR}. 1) Geometrically, A is affine if it contains the line passing through each pair of its points. 2) A singleton, a line, a plane are affine sets 3) an affine set is convex. tvnguyen (University of Science) Convex Optimization 28 / 108 Chapter 2 Topological properties for sets and functions Affine Combination An affine combination of finitely many points x 1 , . . . , x k is defined by k  i=1 α i x i where k  i=1 α i = 1. As for convex sets, we have the following characterization Proposition. A subset A of IR n is affine if and only if it contains every affine combination of finitely many of its points. It is immediate that the translation of an affine set A, namely A + x with x ∈ IR n , is affine. More specifically, the affine sets are just translates of subspaces. tvnguyen (University of Science) Convex Optimization 29 / 108 Chapter 2 Topological properties for sets and functions Linear Subspaces Let us recall that a subspace L is a subset of IR n which satisfies the property : ∀x, y ∈ L, ∀α ∈ IR, x + y ∈ L and αx ∈ L and that two affine sets A and B are parallel if there exists x ∈ IR n such that A = B + x. Proposition. The following statements hold : (i) L is a subspace if and only if L is affine and 0 ∈ L. (ii) Let A be an affine set. Then there exists a unique subspace L parallel to A. Moreover one has A = L + a for every a ∈ A. (iii) The translate of a subspace is an affine set. tvnguyen (University of Science) Convex Optimization 30 / 108 Chapter 2 Topological properties for sets and functions Dimensions and Hyperplanes The dimension of an affine set A is the dimension of its parallel subspace. An affine set of dimension 0 is called a point, an affine set of dimension 1, a line, an affine set of dimension 2, a plane and an affine set of dimension n − 1, an hyperplane. An hyperplane is defined by the set H = {x ∈ IR n | x ∗ , x = b} where x ∗ ∈ IR n , x ∗ = 0, and b ∈ IR. x ∗ is called the normal vector to H. tvnguyen (University of Science) Convex Optimization 31 / 108 Chapter 2 Topological properties for sets and functions Affine Hull Let C be a subset of IR n . The affine hull of C, denoted aff C , is the intersection of all affine subsets of IR n containing C. The affine hull is affine. Proposition. The affine hull of C is the set of all affine combinations of finitely many points of C. Let C = {x 1 , x 2 , . . . , x k }. Then aff C =  k  i=1 α i x i     k  i=1 α i = 1  Examples : aff {x} = {x}, aff [x, y ] is the line generated by x and y. aff B(0, 1) = IR n . tvnguyen (University of Science) Convex Optimization 32 / 108 Chapter 2 Topological properties for sets and functions Affinely Independent Points Definition. Let S = {x 0 , . . . , x k } be a set of k + 1 points of IR n . The points of S are affinely independent if aff S has dimension k Proposition. Let S = {x 0 , . . . , x k } be a set of k + 1 points of IR n . The points of S are linearly independent if and only if the vectors x 1 − x 0 , x 2 − x 0 , . . . , x k − x 0 are linearly independent. tvnguyen (University of Science) Convex Optimization 33 / 108 Chapter 2 Topological properties for sets and functions Relative Interior of a Convex Set Definition. Let C be a nonempty convex subset of IR n . The relative interior of C is the largest open set for the topology induced on aff C that is contained in C. This set is denoted ri C. We have that the relative interior ri C of C is the interior of C for the topology relative to the affine hull of C x ∈ ri C ⇔ x ∈ aff C and ∃δ > 0 such that B(x, δ) ∩ aff C ⊆ C Furthermore, if int C = ∅, then aff C = IR n and ri C = int C Proposition. Let C be a nonempty subset of IR n . Then the relative interior ri C is nonempty. tvnguyen (University of Science) Convex Optimization 34 / 108 Chapter 2 Topological properties for sets and functions Examples 1. Let C = {x}. Then aff C = {x}, int C = ∅ and ri C = {x}. 2. Let C = [a, b] where a, b ∈ IR n with a = b and n ≥ 2. Then aff C is the straight line generated by a and b, int C = ∅ and ri C =]a, b[. 3. Let C = {x ∈ IR n |  n i=1 x i = 1, x i ≥ 0, i = 1, . . . , n}. Then int C = ∅ and aff C = {x ∈ IR n | n  i=1 x i = 1} ri C = {x ∈ IR n | n  i=1 x i = 1, x i > 0, i = 1, . . . , n} For example, in IR 2 , let C be the triangle of vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1). Then the relative interior of the triangle is the interior of the triangle. tvnguyen (University of Science) Convex Optimization 35 / 108 [...]... sets Sr (f ) = {x ∈ R n |f (x) ≤ r } are closed (possibly empty) for all r ∈ IR A function satisfying one of these three properties is also called a closed function tvnguyen (University of Science) Convex Optimization 38 / 108 Chapter 2 Topological properties for sets and functions Illustration tvnguyen (University of Science) Convex Optimization 39 / 108 Chapter 2 Topological properties for sets and. ..Chapter 2 Topological properties for sets and functions Continuity and locally Lipschitz continuity Definition Let f : IRn → IR ∪ {+∞} and x ∈ ri dom f The function f is continuous at x if for all ε > 0 there exists δ > 0 such that ∀y ∈ B(x, δ) ∩ aff domf |f (y ) − f (x)| < ε The function f is locally Lipschitz continuous at x ∈ ri domf if there exists δ > 0 and L > 0 such that ∀y , z... → IR ∪ {+∞} is proper convex, (ii) cl f and f coincide on ri dom f tvnguyen (University of Science) Convex Optimization 40 / 108 Chapter 2 Topological properties for sets and functions Examples Given a nonempty subset C ⊂ IRn , the indicator function δC is defined by 0 if x ∈ C δC (x) = +∞ if not Clearly δC is [closed and] convex if and only if C is [closed and] convex Indeed epi δC = C × IR+ Let (s1... for all x ∈ C by C∞ (x) = {d ∈ IRn : x + t d ∈ C for all t > 0} Proposition The closed convex cone C∞ does not depend on x ∈ C Proposition C∞ = {0} A closed convex set C is compact if and only if tvnguyen (University of Science) Convex Optimization 42 / 108 Chapter 2 Topological properties for sets and functions Illustration tvnguyen (University of Science) Convex Optimization 43 / 108 Chapter 2 Topological. .. (University of Science) Convex Optimization 44 / 108 Chapter 2 Topological properties for sets and functions Property Proposition Let f : IRn → IR ∪ {+∞} be proper l.s.c convex The following statements are equivalent : (i) There is r for which the sublevel set Sr (f ) is nonempty and compact (ii) All the sublevel sets of f are compact (iii) f∞ (d) > 0 for all nonzero d ∈ IRn The sublevel set of f at level... closed and convex function A polyhedral function f is a function whose epigraph is a closed convex polyhedron : epi f = {(x, r ) ∈ IRn × IR | sj , x + αj r ≤ bj for j ∈ J} where J is finite, the (s, α, b)j being given in IRn × IR × IR, (sj , αj ) = 0 Such a function is closed and convex tvnguyen (University of Science) Convex Optimization 41 / 108 Chapter 2 Topological properties for sets and functions. .. Convex Optimization 36 / 108 Chapter 2 Topological properties for sets and functions Proposition Let f : IRn → IR ∪ {+∞} be proper convex Then f is locally Lipschitz continuous on ri dom f In particular f is continuous on ri dom f Corollary Let f : IRn → IR be convex Then f is continuous and locally Lipschitz continuous on IRn Example Any norm on IRn is convex and continuous on IRn tvnguyen (University... of Science) Convex Optimization 37 / 108 Chapter 2 Topological properties for sets and functions Lower semi continuity Definition Let f : IRn → IR ∪ {+∞} be proper f is said to be lower semi continuous (l.s.c) at x ∈ IRn if lim inf f (y ) ≥ f (x) y →x (⇔ ∀{yk } → x lim inf f (yk ) ≥ f (x)) y →x Proposition Let f : IRn → IR ∪ {+∞} The following three properties are equivalent : (i) f is l.s.c on IRn (ii)... / 108 Chapter 2 Topological properties for sets and functions Asymptotic function of a convex function Definition Let f : IRn → IR ∪ {+∞} be proper l.s.c convex The asymptotic cone of epi f is the epigraph of the function f∞ defined by d → f∞ (d) := sup t>0 f (x0 + td) − f (x0 ) f (x0 + td) − f (x0 ) = lim t→+∞ t t where x0 is arbitrary in domf f∞ is proper l.s.c convex and called the asymptotic function... Topological properties for sets and functions Closure or l.s.c hull of a convex function Definition The closure (or l.s.c hull) of a convex function f is the function cl f : IRn → IR ∪ {+∞} defined by epi cl f = cl epi f Proposition The closure of a convex function f : IRn → IR ∪ {+∞} is the supremum of all affine functions minorizing f : cl f (x) = sup { s, x − b : s, y − b ≤ f (y ) for all y ∈ IRn } n (s,b)∈IR . Chapter 2 Topological properties for sets and functions Chapter 2. Topological properties for sets and functions tvnguyen (University. 108 Chapter 2 Topological properties for sets and functions Continuity and locally Lipschitz continuity Definition. Let f : IR n → IR ∪ {+∞} and x ∈ ri dom