VARIABLE OPTIMIZATION APPARATUS, VARIABLE OPTIMIZATION METHOD, AND PROGRAM

Application No. 17/761,592

• Filed: Mar 17, 2022
• Priority: Sep 19, 2019 — nonprovisional of PCTJP2019036692
• Examiner: VILLANUEVA, MARKUS ANTHONY
• Art Unit: 2151
• Tech Center: 2100 — Computer Architecture & Software
• Assignee: Nippon Telegraph and Telephone Corporation
• OA Round: 1 (Non-Final)

Office Action

DETAILED ACTION
Notice of Pre-AIA or AIA Status
The present application, filed on or after March 16, 2013, is being examined under the first inventor to file provisions of the AIA .

Specification
The abstract of the disclosure is objected to because it refers to purported merits. A corrected abstract of the disclosure is required and must be presented on a separate sheet, apart from any other text. See MPEP § 608.01(b).

Claim Rejections - 35 USC § 101
35 U.S.C. 101 reads as follows:
Whoever invents or discovers any new and useful process, machine, manufacture, or composition of matter, or any new and useful improvement thereof, may obtain a patent therefor, subject to the conditions and requirements of this title.

Claims 1-3, 8-10 are rejected under 35 U.S.C. 101 because the claimed invention is directed to a judicial exception (i.e., an abstract idea) without significantly more.
Apparatus claims 1 and 8 will be addressed first, followed by apparatus claims 2 and 9, followed by apparatus claims 3 and 10.

Regarding claim 1, under the Alice Framework Step 1 analysis, the claim falls within the four statutory categories of patentable subject matter: an apparatus.
Under the Alice Framework Step 2A Prong 1 analysis, the claim recites Mathematical Concepts. The claim recites Mathematical Calculations, which is specifically identified as an exemplar in the Mathematical Concepts grouping of abstract ideas:
“
w
∈

R

n

is a variable being an optimization target, and
G
(
w
)

(
=

G

1

(
w
)

+

G

2

(
w
)
)

is a cost function for optimizing the variable
w

, calculated by using input data (note that a function

G

i

(
w
)
:

R

n

→

R
∪
{
∞
}

(
i

=

1
,

2
)

is a closed proper convex function), and

D
:

R

n

→

R

is a strictly convex function (note that the function D is differentiable, and satisfies
∇
D
(
0
)

=

0

), and

R

i

(
i

=

1
,

2
)

and

C

i

(
i

=

1
,

2
)

are a D-resolvent operator and a D-Cayley operator defined by following expressions, respectively, [Math. 63]

R

i

=

I
+
(
∇
D

-
1

∘
∂

G

i

)

-
1

C

i

=

I
+

∇
D

-
1

∘
∂

G

i

-
1

∘

I
-

∇
D

-
1

∘
∂

G

i

x

recursively determining a value of the variable w by using the D-resolvent operator

R

i

(
i

=

1
,

2
)

and the D-Cayley operator

C

i

(
i

=

1
,

2
)

,
wherein

x

-

G

i

(
w
)

(
i

=

1
,

2
)

is a strongly convex function approximating the function

G

i

(
w
)

(
i

=

1
,

2
)

, and
wherein the calculating
∇
D
(
w
)

, for a D-resolvent operator

R

1

and a D-Cayley operator

C

1

,

T

1

(
w
)
=

∇

-

G

1

(
w
)
-

∇

-

G

1

(
0
)

is used for calculation of
∇
D
(
w
)

, and for a D-resolvent operator

R

2

and a D-Cayley operator

C

2

, uses

T

2

w

=

∇

-

G

2

w

-

∇

-

G

2

0

.

”

See specification ([0003], [0023], [0127]) describing
w
,

G

w

,

and

G

i

(
w
)

. See specification ([0023-0025], [0035-0037], [0042-0045], [0086-0089], [0091-0094], [0129-0131], [0134], [0161-0165]) describing
D
,

R

i

,

and

C

i

. See specification ([0025], [0048-0050], [0131-0132], [0159]) describing recursively determining. See specification ([0025], [0091-0096], [0130], [0134-0135]) describing

x

-

G

i

(
w
)

. See specification ([0025], [0086-0096], [0131], [0134-0135], [0161-0165], [0194-0198]) describing
∇
D
(
w
)
,

T

1

w

,

and

T

2

(
w
)

. For these reasons, the claim recites Mathematical Concepts.
Under the Alice Framework Step 2A Prong 2 analysis, the claim recites the combination of the following additional elements: a processor and a memory storing instructions configured to execute a method. A processor and a memory storing instructions configured to execute a method are recited at a high level of generality, and are examples of generic computing elements, and/or merely generally linked to a particular technological environment (see MPEP 2106.05(h)(vi): Limiting the abstract idea of collecting information, analyzing it, and displaying certain results of the collection analysis to data related to the electric power grid, because limiting application of the abstract idea to power-grid monitoring is simply an attempt to limit the use of the abstract idea to a particular technological environment). Taken alone or in combination, they fail to integrate the judicial exception into a practical application.
Under the Alice Framework Step 2B Analysis, the additional elements recited above, taken alone or in combination, do not amount to significantly more than the judicial exception. As discussed in the Step 2A Prong 2 Analysis, the claim recites a processor and a memory storing instructions configured to execute a method at a high level of generality, which merely result in “apply it” on a computer, and/or merely generally linking to a particular technologic environment. Since the claim does not include additional elements that, alone or in combination, amount to significantly more than the judicial exception, claim 1 is ineligible.

Under the Alice Framework Step 2A Prong 1 analysis, claim 8 recites Mathematical Concepts. The claim recites Mathematical Calculations, which is specifically identified as an exemplar in the Mathematical Concepts grouping of abstract ideas:
“generating an output image without noise,

based on the recursively determining the value of the variable w upon pixels of an input image
.

”

See specification ([0113-0125], [0187-0190], [0192-0198], [0217-0219]) describing generating an output image. See specification ([0113-0125], [0187-0190], [0192-0198], [0217-0219]) describing recursively determining the value of w. For these reasons, the claim recites Mathematical Concepts.
Under the Alice Framework Step 2A Prong 2 analysis, the claim recites the combination of the following additional elements: for noise elimination. Noise elimination is recited at a high level of generality, and is an example of merely generally linked to a particular technological environment (see MPEP 2106.05(h)(vi): Limiting the abstract idea of collecting information, analyzing it, and displaying certain results of the collection analysis to data related to the electric power grid, because limiting application of the abstract idea to power-grid monitoring is simply an attempt to limit the use of the abstract idea to a particular technological environment). Taken alone or in combination, they fail to integrate the judicial exception into a practical application.
Under the Alice Framework Step 2B Analysis, the additional elements recited above, taken alone or in combination, do not amount to significantly more than the judicial exception. As discussed in the Step 2A Prong 2 Analysis, the claim recites noise elimination at a high level of generality, which merely result in “apply it” on a computer, and/or merely generally linking to a particular technologic environment. Since the claim does not include additional elements that, alone or in combination, amount to significantly more than the judicial exception, claim 8 is ineligible.

Regarding claim 2, under the Alice Framework Step 1 analysis, the claim falls within the four statutory categories of patentable subject matter: an apparatus.
Under the Alice Framework Step 2A Prong 1 analysis, the claim recites Mathematical Concepts. The claim recites Mathematical Calculations, which is specifically identified as an exemplar in the Mathematical Concepts grouping of abstract ideas:
“
w
∈

R

n

is a variable being an optimization target, and
G
(
w
)

(
=

G

1

(
w
)

+

G

2

(
w
)
)

is a cost function for optimizing the variable
w

, calculated by using input data (note that a function

G

i

(
w
)
:

R

n

→

R
∪
{
∞
}

(
i

=

1
,

2
)

is a closed proper convex function),
calculating

w

t
+
1

being (t+1)-th update result of the variable w, wherein x, y, and
z
∈

R

n

are each an auxiliary variable of the variable
w

,
D
:

R

n

→

R

is a strictly convex function (note that the function D is differentiable, and satisfies
∇
D
(
0
)

=

0

),

J

D

is Bregman divergence defined by using the function D,

x

-

G

i

(
w
)

(
i

=

1
,

2
)

is a strongly convex function approximating the function

G

i

(
w
)

(
i

=

1
,

2
)

, and

T

1

(
w
)

and

T

2

(
w
)

are functions defined by following expressions, respectively, [Math. 64]

T

1

w

=
∇

G

1

-

w

-
∇

G

2

-

(
0
)

T

2

w

=
∇

G

1

-

w

-
∇

G

2

-

(
0
)

;
calculating

γ

1

t
+
1

being (t+1)-th update result of a first coefficient

γ

1

by using a following expression,
[Math. 65]

γ

1

t
+
1

=

γ

2

t

T

2

∘

∂

G

1

+
∂

G

2

z

t

2

/

T

1

∘

∂

G

1

+
∂

G

2

z

t

2

calculating

w

t
+
1

being (t+1)-th update result of the variable w by using a following expression,
[Math. 66]

w

t
+
1

=

arg

⁡

min

w

⁡

(

G

1

w

+

J

D

(
w
|
|

z

t

)

)
;

⁡

x

calculating

x

t
+
1

being (t+1)-th update result of the auxiliary variable x by using a following expression,
[Math. 67]

x

t
+
1

=
2

w

t
+
1

-

z

t

;

calculating

γ

2

t
+
1

being (t+1)-th update result of a second coefficient

γ

2

by using a following expression,
[Math. 68]

γ

2

t
+
1

=

γ

1

t
+
1

T

1

∘

∂

G

1

+
∂

G

2

x

t
+
1

2

/

T

2

∘

∂

G

1

+
∂

G

2

x

t
+
1

2

;
calculating

y

t
+
1

being (t+1)-th update result of the auxiliary variable y by using a following expression,
[Math. 69]

y

t
+
1

=

arg

⁡

min

y

⁡

G

2

y

+

J

D

y

|
|
x

t
+
1

;

and
calculating

z

t
+
1

being (t+1)-th update result of the auxiliary variable z by using a following expression,
[Math. 70]

z

t
+
1

=
2

y

t
+
1

-

x

t
+
1

”.

See specification ([0003], [0023], [0127]) describing
w
,

G

w

,

and

G

i

(
w
)

. See specification ([0050-0053], [0086-0089], [0091-0094], [0129-0131], [0134-0137], [0161-0165]) describing

w

t
+
1

. See specification ([0092-0094], [0138-0139], [0167-0168], [0201-0202]) describing

γ

1

t
+
1

. See specification ([0056-0057], [0061-0062], [0066-0067], [0140-0141]) describing

w

t
+
1

by argmin. See specification ([0051-0052], [0107-0108], [0142-0143], [0171-0172], [0207-0208]) describing

x

t
+
1

. See specification ([0092-0094], [0144-0145], [0173-0174], [0207-0208]) describing

γ

2

t
+
1

. See specification ([0051-0052], [0059-0062], [0066-0067], [0146-0147]) describing

y

t
+
1

. See specification ([0051-0052], [0059-0062], [0066-0067], [0148-0150], [0171-0181], [0212-0216]) describing

z

t
+
1

. For these reasons, the claim recites Mathematical Concepts.
Under the Alice Framework Step 2A Prong 2 analysis, the claim recites the combination of the following additional elements: a processor and a memory storing instructions configured to execute a method. A processor and a memory storing instructions configured to execute a method are recited at a high level of generality, and are examples of generic computing elements, and/or merely generally linked to a particular technological environment (see MPEP 2106.05(h)(vi): Limiting the abstract idea of collecting information, analyzing it, and displaying certain results of the collection analysis to data related to the electric power grid, because limiting application of the abstract idea to power-grid monitoring is simply an attempt to limit the use of the abstract idea to a particular technological environment). Taken alone or in combination, they fail to integrate the judicial exception into a practical application.
Under the Alice Framework Step 2B Analysis, the additional elements recited above, taken alone or in combination, do not amount to significantly more than the judicial exception. As discussed in the Step 2A Prong 2 Analysis, the claim recites a processor and a memory storing instructions configured to execute a method at a high level of generality, which merely result in “apply it” on a computer, and/or merely generally linking to a particular technologic environment. Since the claim does not include additional elements that, alone or in combination, amount to significantly more than the judicial exception, claim 2 is ineligible.

Under the Alice Framework Step 2A Prong 1 analysis, claim 9 recites Mathematical Concepts. The claim recites Mathematical Calculations, which is specifically identified as an exemplar in the Mathematical Concepts grouping of abstract ideas:
“generating an output image without noise,

based on pixels of an input image at least using the cost function”

See specification ([0113-0125], [0187-0190], [0192-0198], [0217-0219]) describing generating an output image. See specification ([0091-0092], [0113-0116], [0127], [0130], [0157], [0166], [0187-0188]) describing based on using the cost function. For these reasons, the claim recites Mathematical Concepts.
Under the Alice Framework Step 2A Prong 2 analysis, the claim recites the combination of the following additional elements: for noise elimination. Noise elimination is recited at a high level of generality, and is an example of merely generally linked to a particular technological environment (see MPEP 2106.05(h)(vi): Limiting the abstract idea of collecting information, analyzing it, and displaying certain results of the collection analysis to data related to the electric power grid, because limiting application of the abstract idea to power-grid monitoring is simply an attempt to limit the use of the abstract idea to a particular technological environment). Taken alone or in combination, they fail to integrate the judicial exception into a practical application.
Under the Alice Framework Step 2B Analysis, the additional elements recited above, taken alone or in combination, do not amount to significantly more than the judicial exception. As discussed in the Step 2A Prong 2 Analysis, the claim recites noise elimination at a high level of generality, which merely result in “apply it” on a computer, and/or merely generally linking to a particular technologic environment. Since the claim does not include additional elements that, alone or in combination, amount to significantly more than the judicial exception, claim 9 is ineligible.

Regarding claim 3, under the Alice Framework Step 1 analysis, the claim falls within the four statutory categories of patentable subject matter: an apparatus.
Under the Alice Framework Step 2A Prong 1 analysis, the claim recites Mathematical Concepts. The claim recites Mathematical Calculations, which is specifically identified as an exemplar in the Mathematical Concepts grouping of abstract ideas:
“
w
∈

R

n

is a variable being an optimization target, and
G
(
w
)

(
=

G

1

(
w
)

+

G

2

(
w
)
)

is a cost function for optimizing the variable
w

, calculated by using input data (note that a function

G

i

(
w
)
:

R

n

→

R
∪
{
∞
}

(
i

=

1
,

2
)

is a closed proper convex function),
calculating

w

t
+
1

being (t+1)-th update result of the variable w, wherein x, y, and
z
∈

R

n

are each an auxiliary variable of the variable
w

,
D
:

R

n

→

R

is a strictly convex function (note that the function D is differentiable, and satisfies
∇
D
(
0
)

=

0

),

J

D

is Bregman divergence defined by using the function D,

x

-

G

i

(
w
)

(
i

=

1
,

2
)

is a strongly convex function approximating the function

G

i

(
w
)

(
i

=

1
,

2
)

, and

T

1

(
w
)

and

T

2

(
w
)

are functions defined by following expressions, respectively, [Math. 71]

T

1

w

=
∇

G

1

-

w

-
∇

G

2

-

(
0
)

T

2

w

=
∇

G

1

-

w

-
∇

G

2

-

(
0
)

;
calculating

γ

1

t
+
1

being (t+1)-th update result of a first coefficient

γ

1

by using a following expression,
[Math. 72]

γ

1

t
+
1

=

γ

2

t

T

2

∘

∂

G

1

+
∂

G

2

z

t

2

/

T

1

∘

∂

G

1

+
∂

G

2

z

t

2

calculating

w

t
+
1

being (t+1)-th update result of the variable w by using a following expression,
[Math. 73]

w

t
+
1

=

arg

⁡

min

w

⁡

(

G

1

w

+

J

D

(
w
|
|

z

t

)

)
;

⁡

x

calculating

x

t
+
1

being (t+1)-th update result of the auxiliary variable x by using a following expression,
[Math. 74]

x

t
+
1

=
2

w

t
+
1

-

z

t

;

calculating

γ

2

t
+
1

being (t+1)-th update result of a second coefficient

γ

2

by using a following expression,
[Math. 75]

γ

2

t
+
1

=

γ

1

t
+
1

T

1

∘

∂

G

1

+
∂

G

2

x

t
+
1

2

/

T

2

∘

∂

G

1

+
∂

G

2

x

t
+
1

2

;
calculating

y

t
+
1

being (t+1)-th update result of the auxiliary variable y by using a following expression,
[Math. 76]

y

t
+
1

=

arg

⁡

min

y

⁡

G

2

y

+

J

D

y

|
|
x

t
+
1

;

and
calculating

z

t
+
1

being (t+1)-th update result of the auxiliary variable z by using a following expression,
[Math. 77]

z

t
+
1

=

1
-
α

z

t

-
α
(
2

y

t
+
1

-

x

t
+
1

)

(
α

is a real number satisfying
0
<
α
<
1

)”.

See specification ([0003], [0023], [0127]) describing
w
,

G

w

,

and

G

i

(
w
)

. See specification ([0050-0053], [0086-0089], [0091-0094], [0129-0131], [0134-0137], [0161-0165]) describing

w

t
+
1

. See specification ([0092-0094], [0138-0139], [0167-0168], [0201-0202]) describing

γ

1

t
+
1

. See specification ([0056-0057], [0061-0062], [0066-0067], [0140-0141]) describing

w

t
+
1

by argmin. See specification ([0051-0052], [0107-0108], [0142-0143], [0171-0172], [0207-0208]) describing

x

t
+
1

. See specification ([0092-0094], [0144-0145], [0173-0174], [0207-0208]) describing

γ

2

t
+
1

. See specification ([0051-0052], [0059-0062], [0066-0067], [0146-0147]) describing

y

t
+
1

. See specification ([0066-0067], [0151-0152], [0180-0182], [0212-0216]) describing

z

t
+
1

. For these reasons, the claim recites Mathematical Concepts.
Under the Alice Framework Step 2A Prong 2 analysis, the claim recites the combination of the following additional elements: a processor and a memory storing instructions configured to execute a method. A processor and a memory storing instructions configured to execute a method are recited at a high level of generality, and are examples of generic computing elements, and/or merely generally linked to a particular technological environment (see MPEP 2106.05(h)(vi): Limiting the abstract idea of collecting information, analyzing it, and displaying certain results of the collection analysis to data related to the electric power grid, because limiting application of the abstract idea to power-grid monitoring is simply an attempt to limit the use of the abstract idea to a particular technological environment). Taken alone or in combination, they fail to integrate the judicial exception into a practical application.
Under the Alice Framework Step 2B Analysis, the additional elements recited above, taken alone or in combination, do not amount to significantly more than the judicial exception. As discussed in the Step 2A Prong 2 Analysis, the claim recites a processor and a memory storing instructions configured to execute a method at a high level of generality, which merely result in “apply it” on a computer, and/or merely generally linking to a particular technologic environment. Since the claim does not include additional elements that, alone or in combination, amount to significantly more than the judicial exception, claim 3 is ineligible.

Claim 10 recites similar limitations to claim 9. The claim 9 analysis similarly applies, and claim 10 is equally rejected.

Allowable Subject Matter
Claims 1-3 and 9-10 are rejected, however, would be allowable if rewritten to overcome the respective 35 U.S.C. 101 rejections.
Regarding independent claims 1-3, the prior art of record does not teach or suggest a combination of the entire claim limitations in combination with the dependent limitations. Aspects of the claimed invention were found in the prior art, however, not each limitation in combination as specified.
The following is a statement of reasons for the indication of allowable subject matter:
With respect to claim 1, it is the particular mathematics which are not found in the prior art of record. Specifically, regarding the following limitation:
“[Math. 63]

R

i

=

I
+
(
∇
D

-
1

∘
∂

G

i

)

-
1

C

i

=

I
+

∇
D

-
1

∘
∂

G

i

-
1

∘

I
-

∇
D

-
1

∘
∂

G

i

x

recursively determining a value of the variable w by using the D-resolvent operator

R

i

(
i

=

1
,

2
)

and the D-Cayley operator

C

i

(
i

=

1
,

2
)

,
wherein

x

-

G

i

(
w
)

(
i

=

1
,

2
)

is a strongly convex function approximating the function

G

i

(
w
)

(
i

=

1
,

2
)

, and
wherein the calculating
∇
D
(
w
)

, for a D-resolvent operator

R

1

and a D-Cayley operator

C

1

,

T

1

(
w
)
=

∇

-

G

1

(
w
)
-

∇

-

G

1

(
0
)

is used for calculation of
∇
D
(
w
)

, and for a D-resolvent operator

R

2

and a D-Cayley operator

C

2

, uses

T

2

w

=

∇

-

G

2

w

-

∇

-

G

2

0

”.

With respect to claim 2, it is the particular mathematics which are not found in the prior art of record. Specifically, regarding the following limitation:
“calculating

w

t
+
1

being (t+1)-th update result of the variable w, wherein x, y, and
z
∈

R

n

are each an auxiliary variable of the variable
w

,
D
:

R

n

→

R

is a strictly convex function (note that the function D is differentiable, and satisfies
∇
D
(
0
)

=

0

),

J

D

is Bregman divergence defined by using the function D,

x

-

G

i

(
w
)

(
i

=

1
,

2
)

is a strongly convex function approximating the function

G

i

(
w
)

(
i

=

1
,

2
)

, and

T

1

(
w
)

and

T

2

(
w
)

are functions defined by following expressions, respectively, [Math. 64]

T

1

w

=
∇

G

1

-

w

-
∇

G

2

-

(
0
)

T

2

w

=
∇

G

1

-

w

-
∇

G

2

-

(
0
)

;
calculating

γ

1

t
+
1

being (t+1)-th update result of a first coefficient

γ

1

by using a following expression,
[Math. 65]

γ

1

t
+
1

=

γ

2

t

T

2

∘

∂

G

1

+
∂

G

2

z

t

2

/

T

1

∘

∂

G

1

+
∂

G

2

z

t

2

calculating

w

t
+
1

being (t+1)-th update result of the variable w by using a following expression,
[Math. 66]

w

t
+
1

=

arg

⁡

min

w

⁡

(

G

1

w

+

J

D

(
w
|
|

z

t

)

)
;

⁡

x

calculating

x

t
+
1

being (t+1)-th update result of the auxiliary variable x by using a following expression,
[Math. 67]

x

t
+
1

=
2

w

t
+
1

-

z

t

;

calculating

γ

2

t
+
1

being (t+1)-th update result of a second coefficient

γ

2

by using a following expression,
[Math. 68]

γ

2

t
+
1

=

γ

1

t
+
1

T

1

∘

∂

G

1

+
∂

G

2

x

t
+
1

2

/

T

2

∘

∂

G

1

+
∂

G

2

x

t
+
1

2

;
calculating

y

t
+
1

being (t+1)-th update result of the auxiliary variable y by using a following expression,
[Math. 69]

y

t
+
1

=

arg

⁡

min

y

⁡

G

2

y

+

J

D

y

|
|
x

t
+
1

;

and
calculating

z

t
+
1

being (t+1)-th update result of the auxiliary variable z by using a following expression,
[Math. 70]

z

t
+
1

=
2

y

t
+
1

-

x

t
+
1

”.

With respect to claim 3, it is the particular mathematics which are not found in the prior art of record. Specifically, regarding the following limitation:
“calculating

w

t
+
1

being (t+1)-th update result of the variable w, wherein x, y, and
z
∈

R

n

are each an auxiliary variable of the variable
w

,
D
:

R

n

→

R

is a strictly convex function (note that the function D is differentiable, and satisfies
∇
D
(
0
)

=

0

),

J

D

is Bregman divergence defined by using the function D,

x

-

G

i

(
w
)

(
i

=

1
,

2
)

is a strongly convex function approximating the function

G

i

(
w
)

(
i

=

1
,

2
)

, and

T

1

(
w
)

and

T

2

(
w
)

are functions defined by following expressions, respectively, [Math. 71]

T

1

w

=
∇

G

1

-

w

-
∇

G

2

-

(
0
)

T

2

w

=
∇

G

1

-

w

-
∇

G

2

-

(
0
)

;
calculating

γ

1

t
+
1

being (t+1)-th update result of a first coefficient

γ

1

by using a following expression,
[Math. 72]

γ

1

t
+
1

=

γ

2

t

T

2

∘

∂

G

1

+
∂

G

2

z

t

2

/

T

1

∘

∂

G

1

+
∂

G

2

z

t

2

calculating

w

t
+
1

being (t+1)-th update result of the variable w by using a following expression,
[Math. 73]

w

t
+
1

=

arg

⁡

min

w

⁡

(

G

1

w

+

J

D

(
w
|
|

z

t

)

)
;

⁡

x

calculating

x

t
+
1

being (t+1)-th update result of the auxiliary variable x by using a following expression,
[Math. 74]

x

t
+
1

=
2

w

t
+
1

-

z

t

;

calculating

γ

2

t
+
1

being (t+1)-th update result of a second coefficient

γ

2

by using a following expression,
[Math. 75]

γ

2

t
+
1

=

γ

1

t
+
1

T

1

∘

∂

G

1

+
∂

G

2

x

t
+
1

2

/

T

2

∘

∂

G

1

+
∂

G

2

x

t
+
1

2

;
calculating

y

t
+
1

being (t+1)-th update result of the auxiliary variable y by using a following expression,
[Math. 76]

y

t
+
1

=

arg

⁡

min

y

⁡

G

2

y

+

J

D

y

|
|
x

t
+
1

;

and
calculating

z

t
+
1

being (t+1)-th update result of the auxiliary variable z by using a following expression,
[Math. 77]

z

t
+
1

=

1
-
α

z

t

-
α
(
2

y

t
+
1

-

x

t
+
1

)

(
α

is a real number satisfying
0
<
α
<
1

).

The closest prior art found is Hou, Xiaodong. "Distributed Solutions for a Class of Multi-Agent Optimization Problems". May 2019. Purdue University. West Lafayette, Indiana. (hereinafter “Hou”) which discloses the resolvent and Cayley operators (Pg. 26-27, 3.5 Resolvent and Cayley Operator) and splitting (Pg. 28, Proposition 3.6.1 (Douglas-Rachford Splitting). However, it appears that Hou’s equations are different than what’s claimed. Although they possess some similarities such as the
I

(e.g. Hou’s operators are defined as

R

^

T

=

P
+
T

-
1

P

and

C

^

T

=
2

R

^

T

-
I
d

), they ultimately express different mathematical relationships, and further lack expressing the divergence of function D (
∇
D

) and the composite function.
With respect to claims 2 and 3, which recite similar limitations to each other with the exception of the last clause where claim 2 recites [Math. 70] with accompanying equations and claim 3 recites [Math. 77] with accompanying equations. Claims 2 and 3 recite limitations which reflect the Algorithm 1 Bregman Peaceman-Rachford and Bregman Douglas-Rachford Splitting (Fig. 2). Hou generally teaches the Douglas-Rachford Splitting (Pg. 28, Proposition 3.6.1; Pg. 32-33, 4.2 Generalized Douglas-Rachford Splitting; Pg. 48-53, 5.4 Dual Averaging via Douglas-Rachford Splitting; Pg. 66-67, Proposition 6.3.1). It appears that Hou’s

T

1

,

T

2

,

w

k
+
1

,

and

z

k
+
1

(Pg. 32, Proposition 4.2.1) are defined differently than in the instant application, and it appears Hou does not teach a similar and/or equivalent expression to

γ

1

t
+
1

,

γ

2

t
+
1

,

w

t
+
1

([Math. 66]/[Math. 73])
,

x

t
+
1

,

and

y

t
+
1

.

Ryu, Ernest, and Boyd, Stephen. "A Primer on Monotone Operator Methods Survey". Appl. Comput. Math., V.15, N.1, 2016, pp. 3-43 (hereinafter “Ryu”) discloses the resolvent and Cayley operators (Pg. 19, 6. Resolvent and Cayley Operator; Pg. 21, 6.1. Examples) and splitting (Pg. 28-29, 7.3. Peaceman-Rachford and Douglas-Rachford splitting; Pg. 34, Quasidefinite systems). However, it appears that Ryu’s equations are different than what’s claimed. Although they possess some similarities such as the
I

(e.g. Ryu’s operators are defined as
R
=

I
+
α
A

-
1

and
C
=
2
R
-
I

), they ultimately express different mathematical relationships, and further lack expressing the divergence of function D (
∇
D

) and composite function.
With respect to claims 2 and 3, which recite similar limitations to each other with the exception of the last clauses where claim 2 recites [Math. 70] with accompanying equations and claim 3 recites [Math. 77] with accompanying equations. Claims 2 and 3 recite limitations which reflect the Algorithm 1 Bregman Peaceman-Rachford and Bregman Douglas-Rachford Splitting (Fig. 2). Ryu generally teaches the Peaceman-Rachford Splitting (Pg. 28, Peaceman-Rachford splitting,

x

k
+
1

and

z

k
+
1

) and the Douglas-Rachford Splitting (Pg. 28, Douglas-Rachford splitting,

x

k
+
1

and

z

k
+
1

). It appears that Ryu’s

x

k
+
1

and

z

k
+
1

are defined differently than in the instant application, and it appears Ryu does not teach a similar and/or equivalent expression to

T

1

,

T

2

,

γ

1

t
+
1

,

γ

2

t
+
1

,

w

t
+
1

,

and

y

t
+
1

.

Combettes, Patrick L., "Monotone Operator Theory in Convex Optimization*". North Carolina State University, Department of Mathematics, Raleigh, NC 27695-8205, USA. 1 Jun 2018. (hereinafter “Combettes”) discloses a general overview of convex optimization (Pg. 1, Abstract), resolvent operators (Pg. 2, Theorem 1.3; Pg. 4, 2 Notation and background, Para. 2), and Douglas-Rachford splitting (Pg. 7, Example 3.5; Pg. 14, Douglas-Rachford splitting (4.6); Pg. 15, (4.7)). However, Combettes appears to generally teach these formulations, thereby is silent to disclosing the explicit mathematical relationships claimed.

Davis, Damek., Yin, Wotao., "A Three-Operator Splitting Scheme and its Optimization Applications". Department of Mathematics, University of California, Los Angeles. Los Angeles, CA 90025, USA. 4 Apr 2015. (hereinafter “Davis”) discloses a general overview of operator splitting schemes (Pg. 1, Abstract), resolvent operators (Pg. 2, (1.3)), and Peaceman-Rachford splitting (PRS) (Pg. 3, (1.5)). However, Davis appears to generally teach these formulations, thereby is silent to disclosing the explicit mathematical relationships claimed.

Conclusion
The prior art made of record and not relied upon is considered pertinent to applicant's disclosure. The following are Non-Patent Literature documents co-authored by the named inventor Kenta Niwa:
K. Niwa, H. Chiba, N. Harada, G. Zhang and W. B. Kleijn, "Microphone Array Wiener Post Filtering Using Monotone Operator Splitting," in IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol. 28, pp. 2036-2046, 2020, doi: 10.1109/TASLP.2020.3006342. (hereinafter “Niwa’20”) discloses monotone operator splitting including a cost function, Peaceman-Rachford splitting, and Cayley operators (Pg. 4, Sec. C).

K. Niwa and N. Harada, "Non-negative Matrix Factorization Using Bregman Monotone Operator Splitting," ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Brighton, UK, 2019, pp. 5531-5535, doi: 10.1109/ICASSP.2019.8683509. (hereinafter “Niwa’19”) discloses Bregman monotone operator splitting using the Peaceman-Rachford splitting as an alternative to the Douglas-Rachford splitting (Pg. 5532, 3.1.).

K. Niwa, G. Zhang and W. B. Kleijn, "Fast Edge-consensus Computing Based on Bregman Monotone Operator Splitting," ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Brighton, UK, 2019, pp. 4609-4613, doi: 10.1109/ICASSP.2019.8682333. (hereinafter “Niwa’19-2”) discloses utilizing the Bregman Peaceman-Rachford splitting (Pg. 4610-4611, 3.1. Overview of B-MOS).

K. Niwa, G. Zhang and W. B. Kleijn, "Edge Consensus Computing for Heterogeneous Data Sets," 2018 IEEE Statistical Signal Processing Workshop (SSP), Freiburg im Breisgau, Germany, 2018, pp. 80-84, doi: 10.1109/SSP.2018.8450777. (hereinafter “Niwa’18”) discloses quadratic primal-dual method of multipliers as an advantage over the traditional primal-dual method of multipliers for optimization algorithms using the Peaceman-Rachford splitting (Pg. 81, 2.2.).

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/MARKUS ANTHONY VILLANUEVA/Examiner, Art Unit 2182
/EMILY E LAROCQUE/Primary Examiner, Art Unit 2182

Prosecution Timeline

• Mar 17, 2022: Application Filed
• Aug 18, 2025: Non-Final Rejection mailed — §101 (current)