Convex Convex-Composite FunctionsConvex Convex-Composite Functions Tim Hoheisel (McGill University, Montreal) joint work with James V. Burke (UW, Seattle) Quang V. Nguyen (McGill,

Cone convexity Convex analysis of convex convex-composite functions Applications References

Convex Convex-Composite FunctionsTim Hoheisel (McGill University, Montreal)

joint work with

James V. Burke (UW, Seattle)

Quang V. Nguyen (McGill, Montreal)

West Coast Optimization Meeting

Vancovuer, September 28, 2019


1. Cone convexity


Convex sets and cones

”The great watershed in optimization is not between linearity and nonlinearity, but convexity andnonconvexity.” (R.T. Rockafellar)

In what follows E will be a Euclidean space, i.e. a real-vector space equipped with an inner product〈·, ·〉 : E × E→ R of dimension κ < ∞.

S ⊂ E is said to be

convex if λS + (1 − λ)S ⊂ S (λ ∈ (0, 1));

a cone if λS ⊂ S (λ ≥ 0).

For a cone K its polar is K =v

∣∣∣ 〈v , x〉 ≤ 0 (x ∈ K)

Note that K ⊂ E is a convex cone iff K + K ⊂ K .

0

Figure: Convex set/non-convex cone






convex if λS + (1 − λ)S ⊂ S (λ ∈ (0, 1));



∣∣∣ 〈v , x〉 ≤ 0 (x ∈ K)


0







convex if λS + (1 − λ)S ⊂ S (λ ∈ (0, 1));



∣∣∣ 〈v , x〉 ≤ 0 (x ∈ K)


0



The topology relative to the affine hull

Affine set: A set S = U + x with x ∈ E and a subspace U ⊂ is called affine. This is characterized by

αS + (1 − α)S ⊂ S (α ∈ R).

Affine hull: affM :=⋂S ∈ E | M ⊂ S, S affine .

Relative interior/boundary: C ⊂ E convex.

ri C :=x ∈ C

∣∣∣ ∃ε > 0 : Bε(x) ∩ aff C ⊂ C

(relative interior)x ∈ ri C ⇔ R+(C − x) is a subspace

aff Cri C

C

C aff C ri C

x x x[x, x′] λx + (1 − λ)x′ | λ ∈ R (x, x′)Bε(x) E Bε(x)

Table: Examples for relative interiors




αS + (1 − α)S ⊂ S (α ∈ R).



ri C :=x ∈ C

∣∣∣ ∃ε > 0 : Bε(x) ∩ aff C ⊂ C


aff Cri C

C

C aff C ri C






αS + (1 − α)S ⊂ S (α ∈ R).



ri C :=x ∈ C

∣∣∣ ∃ε > 0 : Bε(x) ∩ aff C ⊂ C

(relative interior)

x ∈ ri C ⇔ R+(C − x) is a subspace

aff Cri C

C

C aff C ri C






αS + (1 − α)S ⊂ S (α ∈ R).



ri C :=x ∈ C

∣∣∣ ∃ε > 0 : Bε(x) ∩ aff C ⊂ C


aff Cri C

C

C aff C ri C






αS + (1 − α)S ⊂ S (α ∈ R).



ri C :=x ∈ C

∣∣∣ ∃ε > 0 : Bε(x) ∩ aff C ⊂ C


aff Cri C

C

C aff C ri C




Cone-induced ordering

Given a cone K ⊂ E, the relation

x ≤K y :⇐⇒ y − x ∈ K (x, y ∈ E)

induces an ordering on E which is a partial ordering if K is convex and pointed1.

Attach to E a largest element +∞• w.r.t. ≤K which satisfies x ≤K +∞• (x ∈ E).

Set E• := E ∪ +∞•.

For F : E1 → E•2 define

dom F :=x ∈ E1

∣∣∣ F(x) ∈ E2

(domain),

gph F :=(x,F(x)) ∈ E1 × E2 | x ∈ dom F

(graph),

rge F :=F(x) ∈ E2 | x ∈ dom F

(range).

1 i.e. K ∩ (−K) = 0




x ≤K y :⇐⇒ y − x ∈ K (x, y ∈ E)



Set E• := E ∪ +∞•.


dom F :=x ∈ E1

∣∣∣ F(x) ∈ E2

(domain),

gph F :=(x,F(x)) ∈ E1 × E2 | x ∈ dom F

(graph),


(range).

1 i.e. K ∩ (−K) = 0




x ≤K y :⇐⇒ y − x ∈ K (x, y ∈ E)



Set E• := E ∪ +∞•.


dom F :=x ∈ E1

∣∣∣ F(x) ∈ E2

(domain),

gph F :=(x,F(x)) ∈ E1 × E2 | x ∈ dom F

(graph),


(range).

1 i.e. K ∩ (−K) = 0




x ≤K y :⇐⇒ y − x ∈ K (x, y ∈ E)



Set E• := E ∪ +∞•.


dom F :=x ∈ E1

∣∣∣ F(x) ∈ E2

(domain),

gph F :=(x,F(x)) ∈ E1 × E2 | x ∈ dom F

(graph),


(range).

1 i.e. K ∩ (−K) = 0


K -convexity

Definition 1 (K -convexity).

Let K ⊂ E2 be a cone and F : E1 → E•2. Then we call F K-convex if

K -epi F :=(x, v) ∈ E1 × E2

∣∣∣ F(x) ≤K v

(K -epigraph)

is convex (in E1 × E2).

F is K -convex ⇐⇒ F(λx + (1 − λ)y) ≤K λF(x) + (1 − λ)F(y) (x, y ∈ E1, λ ∈ [0, 1])

F K -convex, then ri (K -epi F) =(x, v)

∣∣∣ x ∈ ri (dom F), F(x) ri (K) v

K ⊂ L cones: F K -convex ⇒ L -convex

Examples:

K = R+, F : E→ R ∪ +∞ convex

K = Rm+ and F : E→ (Rm)• with Fi : E→ R ∪ +∞ convex (i = 1, . . . ,m)

K =(x, t) ∈ Rn × R | ‖x‖ ≤ t

and F : Rn → Rn × R, F(x) = (x, ‖x‖)

K = Sn+ and F : Sn → (Sn)•,F(X) =

X−1, X 0,

+∞•, else

K = Sn+ and F : Rm×n → Sn , F(X) = XXT

K arbitrary, F affine.


K -convexity



K -epi F :=(x, v) ∈ E1 × E2

∣∣∣ F(x) ≤K v

(K -epigraph)




∣∣∣ x ∈ ri (dom F), F(x) ri (K) v


Examples:

K = R+, F : E→ R ∪ +∞ convex


K =(x, t) ∈ Rn × R | ‖x‖ ≤ t

and F : Rn → Rn × R, F(x) = (x, ‖x‖)

K = Sn+ and F : Sn → (Sn)•,F(X) =

X−1, X 0,

+∞•, else




K -convexity



K -epi F :=(x, v) ∈ E1 × E2

∣∣∣ F(x) ≤K v

(K -epigraph)




∣∣∣ x ∈ ri (dom F), F(x) ri (K) v


Examples:

K = R+, F : E→ R ∪ +∞ convex


K =(x, t) ∈ Rn × R | ‖x‖ ≤ t

and F : Rn → Rn × R, F(x) = (x, ‖x‖)

K = Sn+ and F : Sn → (Sn)•,F(X) =

X−1, X 0,

+∞•, else




K -convexity



K -epi F :=(x, v) ∈ E1 × E2

∣∣∣ F(x) ≤K v

(K -epigraph)




∣∣∣ x ∈ ri (dom F), F(x) ri (K) v


Examples:

K = R+, F : E→ R ∪ +∞ convex


K =(x, t) ∈ Rn × R | ‖x‖ ≤ t

and F : Rn → Rn × R, F(x) = (x, ‖x‖)

K = Sn+ and F : Sn → (Sn)•,F(X) =

X−1, X 0,

+∞•, else




K -convexity



K -epi F :=(x, v) ∈ E1 × E2

∣∣∣ F(x) ≤K v

(K -epigraph)




∣∣∣ x ∈ ri (dom F), F(x) ri (K) v


Examples:

K = R+, F : E→ R ∪ +∞ convex


K =(x, t) ∈ Rn × R | ‖x‖ ≤ t

and F : Rn → Rn × R, F(x) = (x, ‖x‖)

K = Sn+ and F : Sn → (Sn)•,F(X) =

X−1, X 0,

+∞•, else




K -convexity



K -epi F :=(x, v) ∈ E1 × E2

∣∣∣ F(x) ≤K v

(K -epigraph)




∣∣∣ x ∈ ri (dom F), F(x) ri (K) v


Examples:

K = R+, F : E→ R ∪ +∞ convex


K =(x, t) ∈ Rn × R | ‖x‖ ≤ t

and F : Rn → Rn × R, F(x) = (x, ‖x‖)

K = Sn+ and F : Sn → (Sn)•,F(X) =

X−1, X 0,

+∞•, else




K -convexity



K -epi F :=(x, v) ∈ E1 × E2

∣∣∣ F(x) ≤K v

(K -epigraph)




∣∣∣ x ∈ ri (dom F), F(x) ri (K) v


Examples:

K = R+, F : E→ R ∪ +∞ convex


K =(x, t) ∈ Rn × R | ‖x‖ ≤ t

and F : Rn → Rn × R, F(x) = (x, ‖x‖)

K = Sn+ and F : Sn → (Sn)•,F(X) =

X−1, X 0,

+∞•, else




K -convexity



K -epi F :=(x, v) ∈ E1 × E2

∣∣∣ F(x) ≤K v

(K -epigraph)




∣∣∣ x ∈ ri (dom F), F(x) ri (K) v


Examples:

K = R+, F : E→ R ∪ +∞ convex


K =(x, t) ∈ Rn × R | ‖x‖ ≤ t

and F : Rn → Rn × R, F(x) = (x, ‖x‖)

K = Sn+ and F : Sn → (Sn)•,F(X) =

X−1, X 0,

+∞•, else




K -convexity



K -epi F :=(x, v) ∈ E1 × E2

∣∣∣ F(x) ≤K v

(K -epigraph)




∣∣∣ x ∈ ri (dom F), F(x) ri (K) v


Examples:

K = R+, F : E→ R ∪ +∞ convex


K =(x, t) ∈ Rn × R | ‖x‖ ≤ t

and F : Rn → Rn × R, F(x) = (x, ‖x‖)

K = Sn+ and F : Sn → (Sn)•,F(X) =

X−1, X 0,

+∞•, else




K -convexity



K -epi F :=(x, v) ∈ E1 × E2

∣∣∣ F(x) ≤K v

(K -epigraph)




∣∣∣ x ∈ ri (dom F), F(x) ri (K) v


Examples:

K = R+, F : E→ R ∪ +∞ convex


K =(x, t) ∈ Rn × R | ‖x‖ ≤ t

and F : Rn → Rn × R, F(x) = (x, ‖x‖)

K = Sn+ and F : Sn → (Sn)•,F(X) =

X−1, X 0,

+∞•, else




2. Convex analysis of convexconvex-composite functions


Convexity of composite functions

For F : E1 → E•2 and g : E2 → R ∪ +∞ we define

(g F)(x) :=

g(F(x)) if x ∈ dom F ,

+∞ else.

Proposition 2 (Combari et al. ’95/Pennanen ’99/Bot et al. ’08/Burke, H., Nguyen ’19).

Let K ⊂ E2 be a convex cone, F : E1 → E•2 K-convex and g : E2 → R ∪ +∞ convex and proper such

that rge F ∩ dom g , ∅. Ifg(F(x)) ≤ g(y) ((x, y) ∈ K-epi F) (1)

then the following hold:

a) g F is convex and proper.

b) g F is lower semicontinuous under either of the following conditions:

i) g is lsc and F is continuous;ii) 〈v , F〉 is lsc (proper, convex) for all v ∈ −K and g is lsc and K-increasing, i.e.

x ≤K y =⇒ g(x) ≤ g(y) (x, y ∈ E2). (2)

Observe that (1) is strictly weaker than (2)!




(g F)(x) :=


+∞ else.








x ≤K y =⇒ g(x) ≤ g(y) (x, y ∈ E2). (2)





(g F)(x) :=


+∞ else.








x ≤K y =⇒ g(x) ≤ g(y) (x, y ∈ E2). (2)





(g F)(x) :=


+∞ else.







i) g is lsc and F is continuous;

ii) 〈v , F〉 is lsc (proper, convex) for all v ∈ −K and g is lsc and K-increasing, i.e.

x ≤K y =⇒ g(x) ≤ g(y) (x, y ∈ E2). (2)





(g F)(x) :=


+∞ else.








x ≤K y =⇒ g(x) ≤ g(y) (x, y ∈ E2). (2)





(g F)(x) :=


+∞ else.








x ≤K y =⇒ g(x) ≤ g(y) (x, y ∈ E2). (2)



The main result

For φ : E→ R ∪ +∞ and x ∈ E we define

∂φ(x) :=v

∣∣∣ φ(x) + 〈v , x − x〉 ≤ φ(x) (x ∈ dom φ)

(subdifferential);

φ : y ∈ E 7→ supx∈dom φ〈y, x〉 − φ(x) (conjugate).

Theorem 3 (Burke,H., Nguyen ’19).

Let K ⊂ E2 be a closed, convex cone, F : E1 → E•2 K-convex and f : E1 → R ∪ +∞, g : E2 → R ∪ +∞

proper, convex such thatg(F(x)) ≤ g(y) ((x, y) ∈ K-epi F).

andF(ri (dom f) ∩ ri (dom F)) ∩ ri (dom g − K) , ∅. (3)

Then (f + g F is convex and) the following hold:

a) (f + g F)∗(p) = minv∈−K ,y∈E1

g∗(v) + f∗(y) + 〈v , F〉∗ (p − y);

b) ∂(f + g F)(x) = ∂f(x) +⋃

v∈∂g(F(x)) ∂ 〈v , F〉 (x) (x ∈ dom f ∩ dom g F).


The main result


∂φ(x) :=v

∣∣∣ φ(x) + 〈v , x − x〉 ≤ φ(x) (x ∈ dom φ)

(subdifferential);







a) (f + g F)∗(p) = minv∈−K ,y∈E1

g∗(v) + f∗(y) + 〈v , F〉∗ (p − y);

b) ∂(f + g F)(x) = ∂f(x) +⋃



The main result


∂φ(x) :=v

∣∣∣ φ(x) + 〈v , x − x〉 ≤ φ(x) (x ∈ dom φ)

(subdifferential);







a) (f + g F)∗(p) = minv∈−K ,y∈E1

g∗(v) + f∗(y) + 〈v , F〉∗ (p − y);

b) ∂(f + g F)(x) = ∂f(x) +⋃



The main result


∂φ(x) :=v

∣∣∣ φ(x) + 〈v , x − x〉 ≤ φ(x) (x ∈ dom φ)

(subdifferential);







a) (f + g F)∗(p) = minv∈−K ,y∈E1

g∗(v) + f∗(y) + 〈v , F〉∗ (p − y);

b) ∂(f + g F)(x) = ∂f(x) +⋃



The case K = −hzn gFor φ : E→ R ∪ +∞ lsc, proper, convex we define its horizon cone by

hzn φ := v | ∀t ≥ 0 : x + tv ∈ levαφ 2,

where levαφ is any nonempty sublevel set of φ.

Lemma 4 (Burke, H., Nguyen ’19).

Let g : E→ R ∪ +∞ be lsc, proper, convex. Then g is (−hzn g)-increasing.

Corollary 5 (Burke, H., Nguyen ’19).

g : E→ R ∪ +∞ be lsc, proper, convex and let F : E1 → E•2 be (−hzn g)-convex such that

F(ri (dom F)) ∩ ri (dom g) , ∅.

Then(g F)∗(p) = min

v∈E2g∗(v) + 〈v , F〉∗ (p)

and∂(g F)(x) =

⋃v∈∂g(F(x))

∂ 〈v , F〉 (x) (x ∈ dom g F).

Proof.K = −hzn g.

2 I.e. hzn φ is the closed, convex (horizon) cone (levαφ)∞ .



hzn φ := v | ∀t ≥ 0 : x + tv ∈ levαφ 2,








v∈E2g∗(v) + 〈v , F〉∗ (p)

and∂(g F)(x) =

⋃v∈∂g(F(x))

∂ 〈v , F〉 (x) (x ∈ dom g F).

Proof.K = −hzn g.




hzn φ := v | ∀t ≥ 0 : x + tv ∈ levαφ 2,








v∈E2g∗(v) + 〈v , F〉∗ (p)

and∂(g F)(x) =

⋃v∈∂g(F(x))

∂ 〈v , F〉 (x) (x ∈ dom g F).

Proof.K = −hzn g.



Component-wise convex functions

Corollary 6.

Let g : Rm → R ∪ +∞ be proper, convex and Rm+-increasing, F : E→ (Rm)• with Fi (i = 1, . . . ,m)

proper, convex such that

F

m⋂i

ri (dom Fi)

∩ ri (dom g) , ∅.

Then

(g F)∗(p) = minv≥0

g∗(v) +

m∑i=1

viFi

∗ (p)

and

∂(g F)(x) =⋃

v∈∂g(F(x))

m∑i=1

vi∂Fi(x) (x ∈ dom g F).

Proof.Use K = Rm

+ and observe that F is K -convex.


3. Applications


Conic programming

Consider the general conic program

min f(x) s.t. F(x) ∈ −K (4)

with

K ⊂ E2 a closed, convex cone,

F : E1 → E2 K -convex,

f : E1 → R ∪ +∞ proper and convex.

Problem (4) can be written in the additive composite form

minx∈E1

f(x) + (δ−K F)(x).

Then g = δ−K is K -increasing and qualification condition (3) reads

F(ri (dom f)) ∩ ri (−K) , ∅. (5)

Under (5) we haveinf

x∈E1f(x) + (δ−K F)(x) = max

v∈−K−f∗(y) − (δ−K F)∗(−y)

= maxv∈−K

infx∈E1

f(x) +⟨v , F(x)

⟩.


Conic programming


min f(x) s.t. F(x) ∈ −K (4)

with





minx∈E1

f(x) + (δ−K F)(x).


F(ri (dom f)) ∩ ri (−K) , ∅. (5)



v∈−K−f∗(y) − (δ−K F)∗(−y)

= maxv∈−K

infx∈E1

f(x) +⟨v , F(x)

⟩.


Conic programming


min f(x) s.t. F(x) ∈ −K (4)

with





minx∈E1

f(x) + (δ−K F)(x).


F(ri (dom f)) ∩ ri (−K) , ∅. (5)



v∈−K−f∗(y) − (δ−K F)∗(−y)

= maxv∈−K

infx∈E1

f(x) +⟨v , F(x)

⟩.


Conic programming


min f(x) s.t. F(x) ∈ −K (4)

with





minx∈E1

f(x) + (δ−K F)(x).


F(ri (dom f)) ∩ ri (−K) , ∅. (5)



v∈−K−f∗(y) − (δ−K F)∗(−y)

= maxv∈−K

infx∈E1

f(x) +⟨v , F(x)

⟩.


Conic programming II

Consider againmin f(x) s.t. F(x) ∈ −K . (6)

Observe that∂δ−K (y) =

v

∣∣∣ 〈v , y − y〉 ≥ 0 (y ∈ K)

=: N−K (y) (y ∈ −K).

Theorem 7 (Pennanen ’99/Burke, H.,Nguyen ’19).

Let f ∈ Γ(E1), K ⊂ E2 a closed, convex cone, and let F : E1 → E2 be K-convex. Then the condition

0 ∈ ∂f(x) +⋃

v∈N−K (F(x))

∂ 〈v , F〉 (x) (7)

is sufficient for x to be a minimizer of (6). Under the condition F(ri (dom f)) ∩ ri (−K) , ∅ it is alsonecessary.

Corollary 8.

Let f : E1 → R be differentiable and convex, K ⊂ E2 a closed, convex cone, and let F : E1 → E2 bedifferentiable and K-convex. Then the condition

− ∇f(x) ∈ F ′(x)∗N−K (F(x)) (8)

is sufficient for x to be a minimizer of (6). Under the condition rge F ∩ ri (−K) , ∅ it is also necessary.





v

∣∣∣ 〈v , y − y〉 ≥ 0 (y ∈ K)

=: N−K (y) (y ∈ −K).



0 ∈ ∂f(x) +⋃

v∈N−K (F(x))

∂ 〈v , F〉 (x) (7)


Corollary 8.


− ∇f(x) ∈ F ′(x)∗N−K (F(x)) (8)






v

∣∣∣ 〈v , y − y〉 ≥ 0 (y ∈ K)

=: N−K (y) (y ∈ −K).



0 ∈ ∂f(x) +⋃

v∈N−K (F(x))

∂ 〈v , F〉 (x) (7)


Corollary 8.


− ∇f(x) ∈ F ′(x)∗N−K (F(x)) (8)






v

∣∣∣ 〈v , y − y〉 ≥ 0 (y ∈ K)

=: N−K (y) (y ∈ −K).



0 ∈ ∂f(x) +⋃

v∈N−K (F(x))

∂ 〈v , F〉 (x) (7)


Corollary 8.


− ∇f(x) ∈ F ′(x)∗N−K (F(x)) (8)



Variational Gram functionsGiven M ⊂ Sn

+ closed and convex, the associated variational Gram function (VGF) is given by

ΩM : Rn×m → R ∪ +∞, ΩM(X) =12σM(XXT ) := sup

V∈Mtr (VXXT ).

Then

F : X ∈ Rn×m 7→ XXT is Sn+-convex;

σK-epi F (X ,−V) =

12 tr

(XT V†X

), if rge X ⊂ rge V , V 0,

+∞, else(matrix-fractional function);

σM is Sn+-increasing;

M is bounded if and only if rge F ∩ ri (domσM) , ∅.

Proposition 9 (Jalali et al. ’17/Burke, Gao, H.’ 18/Burke, H.,Nguyen ’19).

Let M ⊂ Sn+ be nonempty, convex and compact. Then Ω∗M is finite-valued and given by

Ω∗M(X) =12

minV∈M

tr (XT V†X)

∣∣∣ rge X ⊂ rge V.

and

∂ΩM(X) =

VX

∣∣∣∣∣∣ V ∈ arg maxM

⟨XXT , ·

⟩ is compact for all X.





V∈Mtr (VXXT ).

Then



12 tr

(XT V†X







Ω∗M(X) =12

minV∈M

tr (XT V†X)


and

∂ΩM(X) =

VX


⟨XXT , ·






V∈Mtr (VXXT ).

Then



12 tr

(XT V†X







Ω∗M(X) =12

minV∈M

tr (XT V†X)


and

∂ΩM(X) =

VX


⟨XXT , ·






V∈Mtr (VXXT ).

Then



12 tr

(XT V†X







Ω∗M(X) =12

minV∈M

tr (XT V†X)


and

∂ΩM(X) =

VX


⟨XXT , ·






V∈Mtr (VXXT ).

Then



12 tr

(XT V†X







Ω∗M(X) =12

minV∈M

tr (XT V†X)


and

∂ΩM(X) =

VX


⟨XXT , ·






V∈Mtr (VXXT ).

Then



12 tr

(XT V†X







Ω∗M(X) =12

minV∈M

tr (XT V†X)


and

∂ΩM(X) =

VX


⟨XXT , ·



Extending the matrix-fractional function

Consider the Euclidean space G := Cn×m × Hn equipped with the inner product

〈·, ·〉 : ((X ,U), (Y ,V)) ∈ G × G 7→ Re tr (Y∗X) + Re tr (VU).

Define the mapping F : G→ (Hn)• by

F(X ,V) :=

X∗V†X if rge X ⊂ rge V ,

+∞•, else, (9)

where X∗ is the adjoint of X and V† is the Moore-Penrose pseudoinverse of V .

Proposition 10 (Burke, H., Nguyen, ’11).

Let F : G→ (Hn)• be given by (9) and define γ : G→ R ∪ +∞ by

γ(X ,V) :=

12 tr (F(X ,V)) , if rge X ⊂ rge V , V 0

+∞, else.

Then γ is lsc, proper, and convex, hence a support function.






F(X ,V) :=


+∞•, else, (9)




γ(X ,V) :=


+∞, else.







F(X ,V) :=


+∞•, else, (9)




γ(X ,V) :=


+∞, else.



Spectral FunctionsConsider the function

F : Sn → Rn , F(X) = (λ1(X), . . . , λn(X)), (X ∈ Sn) (10)

where λ1(X) ≥ . . . ≥ λn(X) are the eigenvalues of X .

Let

K =

v ∈ Rn

∣∣∣∣∣∣∣k∑

i=1

vi ≥ 0, k = 1, . . . , n − 1,n∑

i=1

vi = 0

.Then

F is K -convex;

dom F = Sn ;

If g : Rn → R ∪ +∞ is proper, convex and permutation invariant3, theng(F(X)) ≤ g(y) ((X , y) ∈ K -epi F).

Proposition 11 (Lewis ’95/Burke, H., Nguyen ’19).

Let g be proper, convex and permutation invariant and let F : Sn → Rn be given by (10). Then thefollowing hold:

a) g F is convex and (g F)∗ = g∗ F .

b) For all X ∈ F−1(dom g) we have that

∂(g F)(X) =⋃

v∈∂g(λ(X))

convU∗diag(v)U

∣∣∣ U∗XU = diag(F(X)), U∗U = In.

3g(Py) = g(y) for any permuation matrix P.




where λ1(X) ≥ . . . ≥ λn(X) are the eigenvalues of X .Let

K =

v ∈ Rn

∣∣∣∣∣∣∣k∑

i=1

vi ≥ 0, k = 1, . . . , n − 1,n∑

i=1

vi = 0

.

Then

F is K -convex;

dom F = Sn ;






∂(g F)(X) =⋃

v∈∂g(λ(X))

convU∗diag(v)U







K =

v ∈ Rn

∣∣∣∣∣∣∣k∑

i=1

vi ≥ 0, k = 1, . . . , n − 1,n∑

i=1

vi = 0

.Then

F is K -convex;

dom F = Sn ;






∂(g F)(X) =⋃

v∈∂g(λ(X))

convU∗diag(v)U







K =

v ∈ Rn

∣∣∣∣∣∣∣k∑

i=1

vi ≥ 0, k = 1, . . . , n − 1,n∑

i=1

vi = 0

.Then

F is K -convex;

dom F = Sn ;






∂(g F)(X) =⋃

v∈∂g(λ(X))

convU∗diag(v)U







K =

v ∈ Rn

∣∣∣∣∣∣∣k∑

i=1

vi ≥ 0, k = 1, . . . , n − 1,n∑

i=1

vi = 0

.Then

F is K -convex;

dom F = Sn ;






∂(g F)(X) =⋃

v∈∂g(λ(X))

convU∗diag(v)U







K =

v ∈ Rn

∣∣∣∣∣∣∣k∑

i=1

vi ≥ 0, k = 1, . . . , n − 1,n∑

i=1

vi = 0

.Then

F is K -convex;

dom F = Sn ;






∂(g F)(X) =⋃

v∈∂g(λ(X))

convU∗diag(v)U







K =

v ∈ Rn

∣∣∣∣∣∣∣k∑

i=1

vi ≥ 0, k = 1, . . . , n − 1,n∑

i=1

vi = 0

.Then

F is K -convex;

dom F = Sn ;






∂(g F)(X) =⋃

v∈∂g(λ(X))

convU∗diag(v)U




4. References


J. M. Borwein: Optimization with respect to Partial Orderings. Ph.D. Thesis, University of Oxford,1974.

R.I. Bot, S.-M. Grad, and G. Wanka: A new constraint qualification for the formula of thesubdifferential of composed convex functions in infinite dimensional spaces. MathematischeNachrichten 281, 2008, pp. 1088–1107.

R.I. Bot, S.-M. Grad, and G. Wanka: Generalized Moreau-Rockafellar results for composed convexfunctions. Optimization 58(7), 2009, pp. 917–933.

J. V. Burke and T. Hoheisel: Matrix support functionals for inverse problems, regularization, andlearning. SIAM Journal on Optimization 25, 2015, pp. 1135–1159.

J. V. Burke, Y. Gao and T. Hoheisel: Convex Geometry of the Generalized Matrix-FractionalFunction. SIAM Journal on Optimization 28, 2018, pp. 2189–2200.

J. V. Burke, Y. Gao, and T. Hoheisel: Variational properties of matrix functions via the generalizedmatrix-fractional function. SIAM Journal on Optimization, to appear.

J. V. Burke, T. Hoheisel, and Q.V. Nguyen:A study of convex convex-composite functions via infimalconvolution with applications. arXiv:1907.08318.

A. Jalali, M. Fazel, and L. Xiao: Variational Gram Functions: Convex Analysis and Optimization.SIAM Journal on Optimization 27(4), 2017, pp. 2634–2661.

J.-B. Hiriart-Urruty: A Note on the Legendre-Fenchel Transform of Convex Composite Functions.in Nonsmooth Mechanics and Analysis. Eds. P. Alart, O. Maisonneuve, and R. T. Rockafellar,Springer, 2006, pp. 35–46.


A.G. Kusraev and S.S. Kutateladze: Subdifferentials: theory and applications. Mathematics and itsApplications, 323. Kluwer Academic Publishers Group, Dordrecht, 1995.

A.S. Lewis: The convex analysis of unitarily invariant matrix functions. Journal of Convex Analysis2(1–2), 1995, pp. 173–183.

A.S. Lewis:Convex analysis on the hermitian Matrices SIAM Journal on Optimimization 6(1), 1996,pp. 164–177.

T. Pennanen: Graph-Convex Mappings and K-Convex Functions. Journal of Convex Analysis 6(2),1999, pp. 235–266.

R.T. Rockafellar: Convex Analysis. Princeton Mathematical Series, No. 28. Princeton UniversityPress, Princeton, N.J. 1970.

R.T. Rockafellar and R.J.-B. Wets: Variational Analysis. Grundlehren der MathematischenWissenschaften, Vol. 317, Springer-Verlag, Berlin, 1998.

Documents

Convex Convex-Composite FunctionsConvex Convex-Composite Functions Tim Hoheisel (McGill University, Montreal) joint work with James V. Burke (UW, Seattle) Quang V. Nguyen (McGill,