This document presents a structured composite minimization problem involving minimizing the sum of functions over direct Hilbert sums. It gives examples of how standard convex composite problems and regularization problems can be represented as special cases of this structured problem. It discusses options for activating the functions using proximal operators or separately handling the linear operators and functions. It poses two open problems - developing an efficient distributed block iterative method to solve the structured problem and extending it to maximally monotone inclusions.

1. Structured composite problem in direct
Hilbert sums
Farid Benmouffok
Department of Mathematics,
North Carolina State University
Advisor: Patrick L. Combettes
March 21, 2018
Farid Benmouffok Structured composite problem in direct Hilbert sums 1/7

2. Standard convex composite problem
Let H is a real Hilbert space and let (Gi)i∈I is a finite family of
real Hilbert spaces.
For every i ∈ I, Li is a nonzero bounded linear operator from
H to Gi, and gi ∈ Γ0(Gi) .
Problem 1 is to
minimize
x∈H
i∈I
gi(Lix). (1)
Farid Benmouffok Structured composite problem in direct Hilbert sums 2/7

3. Structured problem
Each gi has a decomposition gi = k∈Ki
gik and each
Li : x → (Lik x)k∈Ki
, where:
Ki ⊂ N
Gi as Hilbert sum decomposition k∈Ki
Gik
Lik : H → Gik is linear and bounded
gik ∈ Γ0(Gik )
If cardKi = +∞, (∀k ∈ Ki) gik gik (0) = 0.
Problem 2 is to
minimize
x∈H
i∈I k∈Ki
gik (Lik x). (2)
Farid Benmouffok Structured composite problem in direct Hilbert sums 3/7

4. Example 1
minimize
x∈H
1
2
Tx − z 2
+
k∈N
φk ( x | ek ). (3)
Farid Benmouffok Structured composite problem in direct Hilbert sums 4/7

5. Example 1
minimize
x∈H
1
2
Tx − z 2
+
k∈N
φk ( x | ek ). (3)
This is a special case of Problem 2 with
I = {1,2}
K1 = {1}, g1,1 = 1
2
· −z 2, and L1 = T



K2 = N
G2,k = R, G2 = k∈K2
R = 2(K2)
g2,k = φk , and L2 : x → ( x | ek )k∈K2
Farid Benmouffok Structured composite problem in direct Hilbert sums 4/7

6. Example 2
minimize
x∈RN
1
2
Tx − z 2
+
M
k=1
x|Gk 2 where, M N (4)
Farid Benmouffok Structured composite problem in direct Hilbert sums 5/7

7. Example 2
minimize
x∈RN
1
2
Tx − z 2
+
M
k=1
x|Gk 2 where, M N (4)
This is a special case of Problem 2 with
I = {1,2}
K1 = {1}, g1,1 = 1
2
· −z 2, and L1 = T



K2 = {1,...,M}
G2,k = R|Gk |, G2 = k∈K2
G2,k = RN
g2,k = · 2 and L2 : x → ( x | ei )i∈Gk
k∈K2
Farid Benmouffok Structured composite problem in direct Hilbert sums 5/7

8. Example 3
minimize
x∈H1
0
(Ω)
p
i=1
α Tix − zi Gi
+
k∈N
β| x|ek | +
1
2
| x|ek |2
+ tv(x). (5)
Farid Benmouffok Structured composite problem in direct Hilbert sums 6/7

9. Example 3
minimize
x∈H1
0
(Ω)
p
i=1
α Tix − zi Gi
+
k∈N
β| x|ek | +
1
2
| x|ek |2
+ tv(x). (5)
This is a special case of Problem 2 with
I = {1,··· ,p + 2} and H = H1
0
(Ω)
Ki = {1}, gi1 = α · −zi Gi
and Li = Ti, if 1 i p
Kp+1 = N, Gp+1 = 2(N), gp+1,k = β · 1 + 1
2
· 2
2
, and
Lp+1 : x → ( x | ek )k∈N
Kp+2 = {1}, Gp+2 = L2(Ω) ⊕ L2(Ω), gp+2,1 : y → Ω
|y(ω)|2dω,
and Lp+2 =
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10. Open problems
A prevalent viewpoint in modern proximal splitting algorithms is
that to activate each function gik appearing in the model there
are two options:
If gik is smooth, i.e, real-valued and differentiable
everywhere with a Lipschitzian gradient, then use gik ;
otherwise, use gik ◦ Lik proximally, i.e, via its proximity
operator
If Lik L∗
ik = αId, then we have an explicit expression of
the proximity operator of gik ◦ Lik ;
otherwise, we can use the splitting algorithm to gik
and Lik separately, [Combettes and Eckstein 2018].
Question 1: Develop an efficient distributed block iterative
method to solve the Problem 2
Question 2: Extention to maximally monotone inclusion of
the form
0 ∈
i∈I k∈Ki
L∗
ik ◦ Aik ◦ Lik (6)
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