Definition of convex function in a vector space

Transcription

1 Convex functions Nonlinear optimization Instituto Superior Técnico and Carnegie Mellon University PhD course João Xavier TAs: Brian Swenson, Shanghang Zhang, Lucas Balthazar

2 Convex functions

3 Nonconvex functions

4 Definition of convex function in a vector space Definition. A function f : V R {+ } in a vector space V is convex if f + and f ((1 α)x + αy) (1 α)f(x) + αf(y) for all x, y S and α [0, 1]. (0(+ ) := 0) Definition. The domain of f is dom f = {x V : f(x) < + }. If f is convex, its domain is a convex set

5 Convexity is a 1D property Proposition. f is convex if and only φ : R R {+ }, φ(t) = f (p + td), is convex for any p and d in V. f p d

6 How do we recognize convex functions? List of simple ones + Apply convexity-preserving operations

7 Simple convex functions affine norms indicators

8 Affine function in R f(x) =sx + r r s x

9 Affine function in R 2 f(x) =s T x + r r x 2 x 1

10 Affine function Definition. An affine function f : V R is a map of the form f(v) = l(v) + r for some linear function l : V R and some r R. Examples: f : R n R, f(x) = s T x + r (s R n, r R) f : R n m R, f(x) = tr ( S T X ) + r (S R n m, r R) f : S n R, f(x) = tr (SX) + r (S S n, r R) Theorem. An affine function is convex.

11 Example: network flow x 1 x 4 c 1 c 4 x 3 1 s t 1 c 5 x 2 x 5 c 2 c 3 Formulation that minimizes cost: minimize x 1,x 2,x 3,x 4,x 5 c 1 x 1 + c 2 x 2 + c 3 x 3 + c 4 x 4 + c 5 x 5 subject to 1 = x 1 + x 2 x 1 + x 3 = x 4 x 2 = x 3 + x 5 x 4 + x 5 = 1 x 1, x 2, x 3, x 4, x 5 0

12 Norm in R f(x) = x x

13 Norm in R 2 f(x) =kxk 2 x 2 x 1

14 Theorem. A norm is a convex function. Examples: f : R n R, f(x) = x 2 f : R n R, f(x) = x f : S n R, f(x) = X F

15 Indicators Definition. The indicator of a set S V is the function { 0, if x S i S : V R {+ }, i S (x) = +, otherwise S Theorem. The indicator of a convex set is a convex function.

16 Indicators allow to pass constraints to the objective: is equivalent to minimize x subject to f(x) x A x B is equivalent to minimize x subject to f(x) + i B (x) x A is equivalent to minimize x f(x) + i A (x) + i B (x) minimize x f(x) + i A B (x)

17 Convexity through differentiability Theorem (1st order criterion). Let S be an open convex subset of R n and f : S R be a differentiable function. Then, f is convex f(x) f(y) + f(y) T (x y) for all x, y S. f(x) f(y)+rf(y) T (x y) y x Mostly useful in the direction: offers affine lower bounds to convex functions produces interesting inequalities

18 Convexity through differentiability Theorem (2nd order criterion). Let S be an open convex subset of R n and f : S R be a twice-differentiable function. Then, f is convex 2 f(y) 0 for all y S. f(x) y x Mostly useful in the direction: proves f is convex Commonly, the 2nd order Taylor expansion f(y) + f(y) T (x y) (y x)t 2 f(y)(y x) is not an upper bound on f(x)

19 Examples f : R ++ R, f(x) = 1 x f : R ++ R, f(x) = log(x) f : R + R, f(x) = x log(x) with 0 log(0) := 0 f : R n R, f(x) = x T Ax + b T x + c, with A symmetric, is convex iff A 0 f : R n R, f(x 1,..., x n ) = log (e x1 + + e xn ) f : R n R ++ R, f(x, y) = xt x y

20 Theorem. For any A S n ++ and B S n there exists S R n n such that A = SS T and B = SΛS T where Λ R n n is diagonal, with the eigenvalues of A 1/2 BA 1/2. Further examples of convex functions: f : S n ++ R, f(x) = tr ( X 1) f : S n ++ R, f(x) = log det(x)

21 Operations that preserve convexity conic combination composition with affine map pointwise supremum

22 Conic combination preserves convexity Theorem. Let f i : V R {+ } be convex functions and α i 0 for i = 1,..., n. If n i=1 dom f i, then is convex. f = α 1 f α n f n Example: basis pursuit with denoising minimize x Ax b ρ x 1 }{{} f(x) for given A R m n, b R m, and ρ > 0

23 Composition with affine map preserves convexity Theorem. Let A : V W be an affine map, f : W R {+ } be convex, and A(V ) dom f. Then is convex. f A : V R {+ } Example: logistic regression (x k, y k ) R n {0, 1}, k = 1,..., K, is the labeled training set label is generated randomly from feature vector P(Y = y X = x, s, r) = ey(st x+r) 1 + e st x+r what should be the classifier parameters (s, r) R n R? K maximize s,r k=1 log P (Y = y k X = x k, s, r)

24 Pointwise supremum preserves convexity Theorem. Let f i : V R {+ }, i I, be a family a convex functions and suppose that f = sup i I f i +. Then, f : V R {+ } is convex. Example: fire-station placement p 1,..., p K are the locations of villages what should be the position x of the fire-station? minimize x max { x p 1,, x p K } }{{} f(x)

25 Example: maximum eigenvalue function λ max : S n R X λ max (X) does not have a closed-form expression variational characterization from linear algebra λ max (X) = sup{q T Xq : q = 1} offers the representation λ max = sup {f q (X) : q Q} with f q : S n R, f q (X) = q T Xq and Q = {q R n : q = 1} f q is a convex function (linear function of X) we conclude λ max is a convex function