Convex functions and convex cones on the Euclidean Spacecmcs.fzu.edu.cn/conf/opt2016/Yulin Zhang.pdfYu-Lin Chang Department of Mathematics National Taiwan Normal University 11 19-21,

MotivationPrevious ResultsRecent Results

Convex functions and convex cones on theEuclidean Space

Yu-Lin Chang

Department of MathematicsNational Taiwan Normal University

11 19-21, 2016

Yu-Lin Chang Convex functions and convex cones on the Euclidean Space


Outline

1 Motivation

2 Previous Results

3 Recent Results



Patr I

Motivation



Monotonicity, Convexity and Cones

Some inequalities are meaningful.For example, arithmetic-geometric inequality is meaningful,Holder’s inequality is meaningful. But why are they meaningful? Iwould say that an inequality is meaningful if it comes from aconvex function. And where does convexity comes from? I wouldguess cones!!!



Second Order Cone

The second-order cone (SOC) in Rn, also called Lorentz cone, isthe set defined as

Kn :={

(x1, x2) ∈ R× Rn−1 | x1 ≥ ‖x2‖}, (1)

where ‖ · ‖ denotes the Euclidean norm. When n = 1, Kn reducesto the set of nonnegative real numbers R+. Kn is also a setcomposed of the squared elements from Jordan algebra (Rn, ◦),where the Jordan product “◦” is a binary operation defined by

x ◦ y := (〈x , y〉, x1y2 + y1x2) (2)

for any x = (x1, x2), y = (y1, y2) ∈ R× Rn−1.



Spectral Decomposition

We recall that each x ∈ Rn admits a spectral decompositionassociated with Kn of the following form

x = λ1(x)u(1)x + λ2(x)u

(2)x , (3)

where λi (x) and u(i)x for i = 1, 2 are the spectral values and the

associated spectral vectors of x , respectively, defined by

λi (x) = x1 + (−1)i‖x2‖, u(i)x =

1

2

(1, (−1)i x2

), (4)

with x2 = x2‖x2‖ if x2 6= 0, and otherwise x2 being any vector in Rn−1

such that ‖x2‖ = 1. The determinant and trace of x are defined asdet(x) := λ1(x)λ2(x) and tr(x) := λ1(x) + λ2(x), respectively.



Inequalities I on SOC

Theorem

The following functions associated with Kn ⊆ Rn are all strictlyconvex with e = (1, 0).

(a) F1(x) = − ln(det(x)) for x ∈ intKn.

(b) F2(x) = tr(x−1) for x ∈ intKn.

(c) F3(x) = tr(h(x)) for x ∈ intKn, where

h(x) =

{xp+1−ep+1 + x1−q−e

q−1 if p ∈ [0, 1], q > 1;xp+1−ep+1 − ln x if p ∈ [0, 1], q = 1.



Inequalities II on SOC

Theorem (Cont.)

The following functions associated with Kn ⊆ Rn are all strictlyconvex with e = (1, 0).

(d) F4(x) = − ln(det(e − x)) for x ≺Kn e.

(e) F5(x) = tr((e − x)−1 ◦ x) for x ≺Kn e.

(f) F6(x) = tr(exp(x)) for x ∈ Rn.

(g) F7(x) = ln(det(e + exp(x))) for x ∈ Rn.

(h) F8(x) = tr

(x + (x2 + 4e)1/2

2

)for x ∈ Rn.



Inequalities III on SOC

Theorem (Cont.)

For any x ∈ K, let F9(x) := −[det(x)]p with 0 < p < 1. Then,

(a) F9 is twice continuously differentiable in intK.

(b) F9 is convex when p ≤ 12 , and moreover, it is strictly

convex when p < 12 in intK.



Inequalities IV on SOC

Theorem (Cont.)

Suppose x ∈ Kn and y ∈ Kn. Then the following inequalities hold.

(a) det(αx + (1− α)y) ≥ (det(x))α(det(y))1−α for any0 < α < 1.

(b) det(x + y)1/p ≥ 22p−1 (

det(x)1/p + det(y)1/p)

forany p ≥ 2.

(c) det(x + y)1/2 ≥ det(x)1/2 + det(y)1/2 anddet(x + y) ≥ det(x) + det(y).

(d) det(αx + (1− α)y) ≥ α2det(x) + (1− α)2det(y) forany 0 < α < 1.



Inequalities V on SOC

Theorem (Cont.)

Suppose x ∈ Kn and y ∈ Kn. Then the following inequalities hold.

(e) [det(e + x)]1/2 ≥ 1 + det(x)1/2.

(f) If x �K y , then det(x) > det(y).

(g) tr(x ◦ y) ≥ 2 det(x)1/2det(y)1/2.

(h) det(x)1/2 =

min

{1

2tr(x ◦ y) : det(y) = 1, y �K 0

}.



Question

How can we prove this? Before that let us take a look atsymmetric cone.



Jordan Algebra

Let V be an n-dimensional vector space over the real field R,endowed with a bilinear mapping (x , y) 7→ x ◦ y from V× V intoV. The pair (V, ◦) is called a Jordan algebra if

(i) x ◦ y = y ◦ x for all x , y ∈ V,

(ii) x ◦ (x2 ◦ y) = x2 ◦ (x ◦ y) for all x , y ∈ V.

Note that a Jordan algebra is not necessarily associative, i.e.,x ◦ (y ◦ z) = (x ◦ y) ◦ z may not hold for all x , y , z ∈ V. We call anelement e ∈ V the identity element if x ◦ e = e ◦ x = x for allx ∈ V. A Jordan algebra (V, ◦) with an identity element e is calleda Euclidean Jordan algebra if there is an inner product 〈·, ·〉Vsuch that

(iii) 〈x ◦ y , z〉V = 〈y , x ◦ z〉V for all x , y , z ∈ V.



Symmetric Cones

Given a Euclidean Jordan algebra A = (V, ◦, 〈·, ·〉V), we denote theset of squares as

K :={x2 | x ∈ V

}.

K is a symmetric cone which means that K is a self-dual closedconvex cone with nonempty interior and for any two elementsx , y ∈ intK, there exists an invertible linear transformationT : V→ V such that T (K) = K and T (x) = y .



Rank

For any given x ∈ A, let ζ(x) be the degree of the minimalpolynomial of x , i.e.,

ζ(x) := min{k : {e, x , x2, · · · , xk} are linearly dependent

}.

Then the rank of A is defined as max{ζ(x) : x ∈ V}. In this paper,we use r to denote the rank of the underlying Euclidean Jordanalgebra.



Jordan Frame

Recall that an element c ∈ V is idempotent if c2 = c . Twoidempotents ci and cj are said to be orthogonal if ci ◦ cj = 0. Onesays that {c1, c2, . . . , ck} is a complete system of orthogonalidempotents if

c2j = cj , cj◦ci = 0 if j 6= i for all j , i = 1, 2, · · · , k and∑k

j=1 cj = e.

An idempotent is primitive if it is nonzero and cannot be writtenas the sum of two other nonzero idempotents. We call a completesystem of orthogonal primitive idempotents a Jordan frame. Nowwe state the second version of the spectral decomposition theorem.



Decomposition Theorem

Theorem

Suppose that A is a Euclidean Jordan algebra with rank r . Thenfor any x ∈ V, there exists a Jordan frame {c1, . . . , cr} and realnumbers λ1(x), . . . , λr (x), arranged in the decreasing orderλ1(x) ≥ λ2(x) ≥ · · · ≥ λr (x), such that

x = λ1(x)c1 + λ2(x)c2 + · · ·+ λr (x)cr .

The numbers λj(x) (counting multiplicities), which are uniquelydetermined by x , are called the eigenvalues.



Trace Function and Euclidean Structure

We can define trace function as the following

tr(x) =r∑

j=1

λj(x).

It is shown that 〈x , y〉 = tr(x ◦ y) defines a Euclidean structure onA. We also define determinant function det(x) = λ1λ2 · · ·λr .



Lowner Operator

Suppose φ : R→ R be a scalar valued function and we define theLowner operator associated with φ as

φscV (x) =:r∑

j=1

φ(λj(x))cj ,

where x ∈ V has the spectral decomposition x =∑r

j=1 λj(x)cj .The associated trace function on Lowner Operator is defined asφtrV :=

∑rj=1 φ(λj(x)).



Koranyi’s Results

Theorem

Let x =∑r

j=1 λj(x)cj . If φ is continuously differentiable, then φscVis differentiable at x and its derivative, for any h ∈ V, is given by

(∇φscV

)(x)(h) =

r∑i=1

(φ[1](λ(x))

)iiCii (x)h (5)

+∑

1≤j<l≤r

(φ[1](λ(x))

)jlCjl(x)h

where the coefficient is defined as

φ[1](λ(x))jl :=

{φ′(λj) if λj = λl ,

φ(λj )−φ(λl )λj−λl if λj 6= λl .

(6)



Part II

Previous Results



Differential Properties

Lemma

For any given scalar function φ : J ⊆ R→ R, let φscV : S → V andφtrV : S → R be defined previously. Assume that J is an openinterval in R. Then, the following results hold.

(a) The domain S of φscV and φtrV is open and convex.

(b) If φ is (continuously) differentiable, then φtrV is(continuously) differentiable on S with∇φtrV (x)(h) = 〈h, (φ′)scV (x)〉 for all h ∈ V.

(c) If φ is twice (continuously) differentiable, then φtrV istwice (continuously) differentiable on S with∇2φtrV (x)(h, k) = 〈h,∇(φ

′)scV (x)k〉 for all h, k ∈ V.



Convexity Preservation

Theorem

For any given φ : J ⊆ R→ R, let φscV : S ⊆ V→ Rn andφtrV : S ⊆ V→ R be given previously. Assume that J is an openinterval in R. If φ is twice differentiable on J, then

(a) φ′′

(t) ≥ 0 for any t ∈ J ⇐⇒ ∇2φtrV (x) � 0 for anyx ∈ S ⇐⇒ φtrV is convex in S .

(b) φ′′

(t) > 0 for any t ∈ J ⇐⇒ ∇2φtrV (x) � 0∀x ∈ S =⇒ φtrV is strictly convex in S .



Inequalities on SC I

Theorem

The following functions associated with K are all strictly convex.

(a) F1(x) = − ln(det(x)) for x ∈ intK.

(b) F2(x) = tr(x−1) for x ∈ intK.

(c) F3(x) = tr(h(x)) for x ∈ intK, where

h(x) =

{xp+1−ep+1 + x1−q−e

q−1 if p ∈ [0, 1], q > 1;xp+1−ep+1 − ln x if p ∈ [0, 1], q = 1.

(d) F4(x) = − ln(det(e − x)) for x ≺K e.



Inequalities on SC II

Theorem (Cont.)

The following functions associated with K are all strictly convex.

(e) F5(x) = tr((e − x)−1 ◦ x) for x ≺K e.

(f) F6(x) = tr(exp(x)) for x ∈ V.

(g) F7(x) = ln(det(e + exp(x))) for x ∈ V.

(h) F8(x) = tr

(x + (x2 + 4e)1/2

2

)for x ∈ V.



Inequalities on SC III

Theorem (Cont.)

For any x ∈ K, let F9(x) := −[det(x)]p with 0 < p < 1. Then,

(a) F9 is twice continuously differentiable in intK.

(b) F9 is convex when p ≤ 1r , and moreover, it is strictly

convex when p < 1r in intK where r is the rank of

the Euclidean Jordan Algebra.



Inequalities on SC VI

Theorem (Cont.)

Suppose r is the rank of the Euclidean Jordan Algebra. For anyx �K 0 and y �K 0, the following inequalities hold.

(a) det(αx + (1− α)y) ≥ (det(x))α(det(y))1−α for any0 < α < 1.

(b) det(x + y)1/p ≥ 2rp−1 (

det(x)1/p + det(y)1/p)

forany p ≥ r .

(c) det(x + y)1/r ≥ det(x)1/r + det(y)1/r anddet(x + y) ≥ det(x) + det(y).

(d) det(αx + (1− α)y) ≥ αrdet(x) + (1− α)rdet(y) forany 0 < α < 1.



Inequalities on SC V

Theorem (Cont.)

Suppose r is the rank of the Euclidean Jordan Algebra. For anyx �K 0 and y �K 0, the following inequalities hold.

(e) [det(e + x)]1/r ≥ 1 + det(x)1/r .

(f) If x �K y , then det(x) > det(y).

(g) tr(x ◦ y) ≥ r det(x)1/rdet(y)1/r .

(h) det(x)1/r =

min

{1

rtr(x ◦ y) : det(y) = 1, y �K 0

}.



Proof of III

Proof. (a) Note that det(x) = exp(tr(ln x)) for any x ∈ intK,and tr(ln(x)) = φtrV (x) with φ(t) = ln(t) for t > 0. By Lemmapart(c), tr(ln(x)) is twice continuously differentiable in intK, andhence det(x) is twice continuously differentiable in intK. Sincet1/p with p > 1 is infinite continuously differentiable on (0,+∞),the desired result then follows.



Proof of III

(b) We now prove the convexity over intK. From our constructionwe have that F9(x) = − exp(pφtrV (x)) with φ(t) = ln(t). By directcomputation we get

∇2F9(x)(h, k) = − exp(pφtrV (x))[p2∇φtrV (x)(h)∇φtrV (x)(k)

+p∇2φtrV (x)(h, k)]

From Lemma and Theorem of Koranyi we obtain

∇2φtrV (x) =r∑

i=1

−1

λiλiCii (x) +

∑1≤j<l≤r

−1

λjλlCjl(x)

where λi (1 ≤ i ≤ r) are the eigenvalues of x .



Cases of Symmetric Cones

Theorem

Every simple Euclidean Jordan algebra is isomorphic to one of thefollowing

(i) The Jordan spin algebra Kn.

(ii) The algebra Sn of n × n real symmetric matrices.

(iii) The algebra Hn of all n × n complex Hermitianmatrices.

(iv) The algebra Qn of all n × n quaternion Hermitianmatrices.

(v) The algebra O3 of all 3× 3 octonion Hermitianmatrices.



Part III

Recent Results



What have we just done?(i) Given any one variable convex function f : R→ R.(ii) We construct a convex function F : Rn → R on Rn.(iii) And this construction heavily depends on the symmetric conestructure on the Euclidean spaces. So it is natural to ask that

What would happen tonon-symmetric cones?



An application

Define a function d(x , y) = tr(x + y)− 2tr(P(x1/2)y)1/2. WhereP(x) = 2L2(x)− L(x2) is the quadratic representation. We canshow that

Theorem

For any x , y ∈ K, assume that x and y operator commute. Then(a) d(x , y) ≥ 0.(b) d(x , y) = 0⇐⇒ x = y .(c) d(x , y) = d(y , x).

Moreover d(x , y) is convex.



Circular Cone

An extension of second-order cone (SOC) in Rn, which is calledcircular cone, is the set defined as

Lθ :={

(x1, x2) ∈ R× Rn−1 | x1 ≥ tan θ‖x2‖}, (7)

where ‖ · ‖ denotes the Euclidean norm.



Spectral Decomposition

Similar to second order cone, each x ∈ Rn admits a spectraldecomposition associated with Lθ of the following form

x = λ1(x)u(1)x + λ2(x)u

(2)x , (8)

where λi (x) and u(i)x for i = 1, 2 are the spectral values and the

associated spectral vectors of x . The determinant and trace of xare defined as det(x) := λ1(x)λ2(x) and tr(x) := λ1(x) + λ2(x),respectively.



Convexity Preservation(Cont.)

Theorem

For any given φ : J ⊆ R→ R, let φtrV : S ⊆ V→ R be givenpreviously. Assume that J is an open interval in R. If φ is twicedifferentiable on J, then

(a) φ′′

(t) ≥ 0 and φ′(t) ≥ 0 for any t ∈ J whenπ/4 ≤ θ ≤ pi/2 =⇒ φtrV is convex in S .

(b) φ′′

(t) ≥ 0 and φ′(t) ≤ 0 for any t ∈ J when0 ≤ θ ≤ pi/4 =⇒ φtrV is convex in S .



Thank You!


Documents

Convex functions and convex cones on the Euclidean Spacecmcs.fzu.edu.cn/conf/opt2016/Yulin Zhang.pdfYu-Lin Chang Department of Mathematics National Taiwan Normal University 11 19-21,