Convexity I: Sets and Functions

Transcription

1 Convexity I: Sets and Functions Lecturer: Ryan Tibshirani Convex Optimization / See supplements for reviews of basic real analysis basic multivariate calculus basic linear algebra

2 Last time: why convexity? Why convexity? Simply put: because we can broadly understand and solve convex optimization problems Nonconvex problems are mostly treated on a case by case basis Reminder: a convex optimization problem is of the form min x D f(x) subject to g i (x) 0, i = 1,... m h j (x) = 0, j = 1,... r where f and g i, i = 1,... m are all convex, and h j, j = 1,... r are affine. Special property: any local minimizer is a global minimizer 2

3 Outline Today: Convex sets Examples Key properties Operations preserving convexity Same, for convex functions 3

4 Convex sets Convex set: C R n such that x, y C = tx + (1 t)y C for all 0 t 1 In words, line segment joining any two elements lies entirely in set Convex combination of x 1,... x k R n : any linear combination θ 1 x θ k x k with θ i 0, i = 1,... k, and k i=1 θ i = 1. Convex hull of a set C, conv(c), is all convex combinations of elements. Always convex 4

5 Examples of convex sets Trivial ones: empty set, point, line Norm ball: {x : x r}, for given norm, radius r Hyperplane: {x : a T x = b}, for given a, b Halfspace: {x : a T x b} Affine space: {x : Ax = b}, for given A, b 5

6 Polyhedron: {x : Ax b}, where inequality is interpreted componentwise. Note: the set {x : Ax b, Cx = d} is also a polyhedron (why?) a1 a2 P a5 a3 a4 Simplex: special case of polyhedra, given by conv{x 0,... x k }, where these points are affinely independent. The canonical example is the probability simplex, conv{e 1,... e n } = {w : w 0, 1 T w = 1} 6

7 Cone: C R n such that Cones x C = tx C for all t 0 Convex cone: cone that is also convex, i.e., x 1, x 2 C = t 1 x 1 + t 2 x 2 C for all t 1, t 2 0 x1 x2 0 Conic combination of x 1,... x k R n : any linear combination θ 1 x θ k x k with θ i 0, i = 1,... k. Conic hull collects all conic combinations 7

8 Examples of convex cones Norm cone: {(x, t) : x t}, for a norm. Under l 2 norm 2, called second-order cone Normal cone: given any set C and point x C, we can define N C (x) = {g : g T x g T y, for all y C} This is always a convex cone, regardless of C Positive semidefinite cone: S n + = {X S n : X 0}, where X 0 means that X is positive semidefinite (and S n is the set of n n symmetric matrices) 8

9 Key properties of convex sets Separating hyperplane theorem: two disjoint convex sets have a separating between hyperplane them a T x b a T x b D C a Formally: if C, D are nonempty convex sets with C D =, then there exists a, b such that C {x : a T x b} D {x : a T x b} 9

10 Supporting hyperplane theorem: a boundary point of a convex set has a supporting hyperplane passing through it Formally: if C is a nonempty convex set, and x 0 bd(c), then there exists a such that C {x : a T x a T x 0 } Both of the above theorems (separating and supporting hyperplane theorems) have partial converses; see Section 2.5 of BV 10

11 11 Operations preserving convexity Intersection: the intersection of convex sets is convex Scaling and translation: if C is convex, then is convex for any a, b ac + b = {ax + b : x C} Affine images and preimages: if f(x) = Ax + b and C is convex then f(c) = {f(x) : x C} is convex, and if D is convex then is convex f 1 (D) = {x : f(x) D}

12 12 Example: linear matrix inequality solution set Given A 1,... A k, B S n, a linear matrix inequality is of the form x 1 A 1 + x 2 A x k A k B for a variable x R k. Let s prove the set C of points x that satisfy the above inequality is convex Approach 1: directly verify that x, y C tx + (1 t)y C. This follows by checking that, for any v, v T ( B k (tx i + (1 t)y i )A i )v 0 i=1 Approach 2: let f : R k S n, f(x) = B k i=1 x ia i. Note that C = f 1 (S n +), affine preimage of convex set

13 13 Example: Fantope Given some integer k 0, the Fantope of order k is { } F k = Z S n : 0 Z I, tr(z) = k where recall the trace operator tr(z) = n i=1 Z ii is the sum of the diagonal entries. Let s prove that F k is convex Approach 1: verify that 0 Z, W I and tr(z) = tr(w ) = I implies the same for tz + (1 t)w Approach 2: recognize the fact that F k = {Z S n : Z 0} {Z S n : Z I} {Z S n : tr(z) = k} intersection of linear inequality and equality constraints (hence like a polyhedron but for matrices)

14 14 More operations preserving convexity Perspective images and preimages: the perspective function is P : R n R ++ R n (where R ++ denotes positive reals), P (x, z) = x/z for z > 0. If C dom(p ) is convex then so is P (C), and if D is convex then so is P 1 (D) Linear-fractional images and preimages: the perspective map composed with an affine function, f(x) = Ax + b c T x + d is called a linear-fractional function, defined on c T x + d > 0. If C dom(f) is convex then so if f(c), and if D is convex then so is f 1 (D)

15 15 Example: conditional probability set Let U, V be random variables over {1,... n} and {1,... m}. Let C R nm be a set of joint distributions for U, V, i.e., each p C defines joint probabilities p ij = P(U = i, V = j) Let D R nm contain corresponding conditional distributions, i.e., each q D defines q ij = P(U = i V = j) Assume C is convex. Let s prove that D is convex. Write { D = q R nm p } ij : q ij = n k=1 p, for some p C = f(c) kj where f is a linear-fractional function, hence D is convex

16 16 Convex functions Convex function: f : R n R such that dom(f) R n convex, and f(tx + (1 t)y) tf(x) + (1 t)f(y) for 0 t 1 and all x, y dom(f) (x, f(x)) (y, f(y)) In words, f lies below the line segment joining f(x), f(y) Concave function: opposite inequality above, so that f concave f convex

17 17 Important modifiers: Strictly convex: f ( tx + (1 t)y ) < tf(x) + (1 t)f(y) for x y and 0 < t < 1. In words, f is convex and has greater curvature than a linear function Strongly convex with parameter m > 0: f m 2 x 2 2 is convex. In words, f is at least as convex as a quadratic function Note: strongly convex strictly convex convex (Analogously for concave functions)

18 18 Examples of convex functions Univariate functions: Exponential function: e ax is convex for any a over R Power function: x a is convex for a 1 or a 0 over R + (nonnegative reals) Power function: x a is concave for 0 a 1 over R + Logarithmic function: log x is concave over R++ Affine function: a T x + b is both convex and concave Quadratic function: 1 2 xt Qx + b T x + c is convex provided that Q 0 (positive semidefinite) Least squares loss: y Ax 2 2 is always convex (since AT A is always positive semidefinite)

19 19 Norm: x is convex for any norm; e.g., l p norms, x p = ( n i=1 x p i ) 1/p for p 1, x = max i=1,...n x i and also operator (spectral) and trace (nuclear) norms, X op = σ 1 (X), X tr = r σ r (X) i=1 where σ 1 (X)... σ r (X) 0 are the singular values of the matrix X

20 20 Indicator function: if C is convex, then its indicator function { 0 x C I C (x) = x / C is convex Support function: for any set C (convex or not), its support function IC(x) = max y C xt y is convex Max function: f(x) = max{x 1,... x n } is convex

21 21 Key properties of convex functions A function is convex if and only if its restriction to any line is convex Epigraph characterization: a function f is convex if and only if its epigraph is a convex set epi(f) = {(x, t) dom(f) R : f(x) t} Convex sublevel sets: if f is convex, then its sublevel sets {x dom(f) : f(x) t} are convex, for all t R. The converse is not true

22 22 First-order characterization: if f is differentiable, then f is convex if and only if dom(f) is convex, and f(y) f(x) + f(x) T (y x) for all x, y dom(f). Therefore for a differentiable convex function f(x) = 0 x minimizes f Second-order characterization: if f is twice differentiable, then f is convex if and only if dom(f) is convex, and 2 f(x) 0 for all x dom(f) Jensen s inequality: if f is convex, and X is a random variable supported on dom(f), then f(e[x]) E[f(X)]

23 23 Operations preserving convexity Nonnegative linear combination: f 1,... f m convex implies a 1 f a m f m convex for any a 1,... a m 0 Pointwise maximization: if f s is convex for any s S, then f(x) = max s S f s (x) is convex. Note that the set S here (number of functions f s ) can be infinite Partial minimization: if g(x, y) is convex in x, y, and C is convex, then f(x) = min y C g(x, y) is convex

24 24 Example: distances to a set Let C be an arbitrary set, and consider the maximum distance to C under an arbitrary norm : f(x) = max y C x y Let s check this is convex: f y (x) = x y is convex in x for any fixed y, so by pointwise maximization rule, f is convex Now let C be convex, and consider the minimum distance to C: f(x) = min y C x y Let s check this is convex: g(x, y) = x y is convex in x, y jointly, and C is assumed convex, so apply partial minimization rule

25 25 More operations preserving convexity Affine composition: f convex implies g(x) = f(ax + b) convex General composition: suppose f = h g, where g : R n R, h : R R, f : R n R. Then: f is convex if h is convex and nondecreasing, g is convex f is convex if h is convex and nonincreasing, g is concave f is concave if h is concave and nondecreasing, g concave f is concave if h is concave and nonincreasing, g convex How to remember these? Think of the chain rule when n = 1: f (x) = h (g(x))g (x) 2 + h (g(x))g (x)

26 26 Vector composition: suppose that f(x) = h ( g(x) ) = h ( g 1 (x),... g k (x) ) where g : R n R k, h : R k R, f : R n R. Then: f is convex if h is convex and nondecreasing in each argument, g is convex f is convex if h is convex and nonincreasing in each argument, g is concave f is concave if h is concave and nondecreasing in each argument, g is concave f is concave if h is concave and nonincreasing in each argument, g is convex

27 27 Example: log-sum-exp function Log-sum-exp function: g(x) = log( k i=1 eat i x+b i ), for fixed a i, b i, i = 1,... k. Often called soft max, as it smoothly approximates max i=1,...k (a T i x + b i) How to show convexity? First, note it suffices to prove convexity of f(x) = log( n i=1 ex i ) (affine composition rule) Now use second-order characterization. Calculate i f(x) = 2 ijf(x) = e x i n l=1 ex l e x i e x i e x j n 1{i = j} l=1 ex l ( n l=1 ex l ) 2 Write 2 f(x) = diag(z) zz T, where z i = e x i /( n l=1 ex l). This matrix is diagonally dominant, hence positive semidefinite

28 28 References and further reading S. Boyd and L. Vandenberghe (2004), Convex optimization, Chapters 2 and 3 J.P. Hiriart-Urruty and C. Lemarechal (1993), Fundamentals of convex analysis, Chapters A and B R. T. Rockafellar (1970), Convex analysis, Chapters 1 10,