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Chapter II: Convex Geometry G. Sagnol Convex Optimization: Chapter II - Convex Geometry WS 2017, TU Berlin Chapter II: Convex Geometry 1 Convex, Affine, and Conic hulls We will first introduce the concept of lines, segments, and rays, which play a central role for the definition of affine, convex, and conic sets. The construction of these three notions is similar to the construction of a vector space (which, as you should recall, relies on the possibility to take linear combinations of points P i λixi), except that here we will put some constraints on the coefficients λi, leading to what we call affine, convex, or conic combinations of points. We will present these constructions in parallel, since they only differ by the constraints imposed on the coefficients of the combinations. n Definition 1. (Lines, segments, rays). Let x1; x2 2 R . • The line through x1 and x2 is fθx1 + (1 − θ)x2 : θ 2 Rg. • The segment between x1 and x2 is fθx1 + (1 − θ)x2 : θ 2 [0; 1]g. • The ray through x1 is f θx1 : θ ≥ 0g. Note that the ray through x is in fact the half-line with end-point 0 that goes through x. Definition 2. (Affine, Convex, and Conic sets). Let S be a subset of Rn. • We say that S is an affine set if it contains the line through any 2 points of S: x1; x2 2 S; θ 2 R =) θx1 + (1 − θ)x2 2 S: • We say that S is convex if it contains the segment through any 2 points of S: x1; x2 2 S; θ 2 [0; 1] =) θx1 + (1 − θ)x2 2 S: • We say that S is a cone if it contains the ray through any point of S: x 2 S; θ ≥ 0 =) θx 2 S: By combining the last 2 points of the above definition, it is easy to see that a set C is a convex cone if and only if 8x1; x2 2 C; 8λ1; λ2 2 R+; λ1x1 + λ2x2 2 C: In fact, we can also show that the above definitions can be extended by taking affine (convex, convex conic) combinations of more than just two points. This yields the following Proposition 1. A set S 2 Rn is affine (or convex, or a convex cone) iff it is stable by affine (or convex, or conic) combinations. More precisely, Page 1 of 15 G. Sagnol Convex Optimization: Chapter II - Convex Geometry WS 2017, TU Berlin k Pk Pk • S is affine () 8k 2 N; 8x1;:::; xk 2 S; 8λ 2 R with i=1 λi = 1; i=1 λixi 2 S: k Pk Pk • S is convex () 8k 2 N; 8x1;:::; xk 2 S; 8λ 2 R+ with i=1 λi = 1; i=1 λixi 2 S: k Pk • S is a convex cone () 8k 2 N; 8x1;:::; xk 2 S; 8λ 2 R+; i=1 λixi 2 S: Proof. We only handle the \convex" case, the other two are similiar. The implication (= is just the case k = 2. For =) , we proceed by induction on k 2 N. The case k = 1 is clear. Now, let S be a convex set, and assume that n k+1 Pk+1 the induction hypothesis is true up to order k. Let x1;:::; xk+1 2 R and let λ 2 R+ satisfy i=1 λi = 1. We can write k+1 k X X λixi λixi = (1 − λk+1) +λk+1xk+1 1 − λk+1 i=1 i=1 | {z } x0 λi Now, observe that the coefficients µi = (i = 1; : : : ; k) sum to 1, so by the induction hypothesis we have 1−λk+1 Pk+1 x0 2 S. It follows that i=1 λixi = (1 − λk+1)x0 + λk+1xk+1 2 S by convexity of S. Pk Note: The vector i=1 λixi is called an affine combination of the xi's when the λi's sum to one, a convex combination of the xi's when the λi's are nonnegative and sum to one, and a conic combination when the λi's are nonnegative (without restriction on their sum). Now we can define the notions of affine, convex, and conic hull of a set S, which correspond to the smallest (in the sense of inclusion) affine set (convex set, convex cone) that contain S. In other words, the affine (convex, conic) hull of S is the intersection of all affine (convex, convex conic) supersets of S. This definition can be useful, but is not really practical, since we are talking about taking the intersection of an infinite family of supersets of S. Fortunately, there is an equivalent definition, which gives an explicit construction: the hulls are obtained by taking all possible affine (convex, conic) combinations of points in S. Definition 3. (Affine, Convex, and Conic hull). Let S be a subset of Rn. • The vector space spanned by S is: k X k span S := f λixi : k 2 N; 8i 2 [k]; xi 2 S; λ 2 R g: i=1 • The affine hull of S is: k X k T aff S := f λixi : k 2 N; 8i 2 [k]; xi 2 S; λ 2 R ; 1 λ = 1g: i=1 • The convex hull of S is: k X k T conv S := f λixi : k 2 N; 8i 2 [k]; xi 2 S; λ 2 R+; 1 λ = 1g: i=1 • The conic hull of S is: k X k cone S := f λixi : k 2 N; 8i 2 [k]; xi 2 S; λ 2 R+g: i=1 Proof. There is nothing to show for span , the definition is just given here for the sake of completeness, and to emphasize the parallel with the three hull-definitions. Page 2 of 15 G. Sagnol Convex Optimization: Chapter II - Convex Geometry WS 2017, TU Berlin We need to show that the above statements are indeed equivalent to the original definition of the different hulls (as intersection of all affine/convex/convex conic supersets). We'll do this for the convex case only, the other two are similar. Let R denote the set on the right hand side of the definition, and let S0 denote an arbitrary convex superset P P 0 0 0 of S. For any element x = i λixi of R (with i λi = 1), we have that xi 2 S , so by convexity of S it holds x 2 S . This shows R ⊆ S0, and hence R ⊆ conv S because S0 was arbitrary. For the converse inclusion, it is sufficient to show that R is a convex superset of S. For all x 2 S we can write x as a convex combination of itself (i.e., with P P k = 1), so S ⊆ R. It remains to show that R is convex. To do so, let x = i λixi 2 R and y = j µj yj 2 R (with P P i λi = j µj = 1). For some α 2 [0; 1] we will show that convex combination z = αx + (1 − α)y is an element of R. Indeed, X X X X z = αλixi + (1 − α)µj yj ; with αλi + (1 − α)µj = α + (1 − α) = 1; i j i j so z is a convex combination of the xi's and the yj 's, and z 2 R. This shows that R is convex, and we have shown conv S ⊆ R. As intuition suggests, an affine set is just a vector space shifted by some constant vector. Indeed, let n Pk k T x1;:::; xk 2 R and consider the affine set L = f i=1 λixi : λ 2 R ; 1 λ = 1g. By using the relation Pk λ1 = 1 − i=2 λi, we can rewrite L as k−1 X k−1 L = fxk + λi(xi − xk): λ 2 R g = xk + span fx1 − xk; x2 − xk;:::; xk−1 − xkg; i=1 where span S denote the vector space spanned by the vectors of S. More generally, the following proposition shows that every affine space can be written as the sum of a subspace and a constant vector (also called offset), or equivalently as the set of solutions of a linear equation: Proposition 2. Let L be an affine space. Then, we have L = x0 + V , where x0 is any element of L, and V is a vector space. The subspace V = L − x0 does not depend on the choice of x0 2 L. Thus, we can define the dimension of L as the dimension of its associated subspace V . If L is an affine space of Rn of dimension m 2 N, then there exists a full-column-rank matrix A 2 Rn×m and a vector b 2 Rn such that m L = fAy + b : y 2 R g: Alternatively, there exists a full-row-rank matrix F 2 Rm×n and a vector g 2 Rm such that n L = fx 2 R : F x = gg: Proof. Let x0 2 L. We first show that V = L − x0 := fx − x0 : x 2 Lg is a vector space. To do this, we select x1; x2 2 L and λ 2 R, we let y1 = x1 − x0 2 V; y2 = x2 − x0 2 V , and we show that λy1 + y2 2 V : λy1 + y2 2 V () λy1 + y2 + x0 2 L () λ(x1 − x0) + (x2 − x0) + x0 2 L () λx1 − λx0 + x2 2 L The last statement is true, because λ − λ + 1 = 1, so λx1 − λx0 + x2 is an affine combination of x0; x1; and x2, and L is stable by affine combinations, cf.
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