Numerical examples - Split generalized equilibrium problem with multiple output sets

3.3 Split generalized equilibrium problem with multiple output sets

3.3.3 Numerical examples

This section provides some examples to illustrate the implementation of our proposed methods, Algorithm 3.3.5. In our experiment, we let g(x) = ^x₃, α_n = _140n+1¹ , δ_n =

3n+14, µ_n = 1−α_n−δ_n, for i = 0,1,2, let r_i = r = 0.5, , β_i,n = ¹₃, γ_n,0 = ¹₂, γ_n,j =

2^j+1, j = 1,2, . . . . Moreover, we consider the effect of varying values of the following parameters θ_i,n = _2+i¹ ,1.0,_3+i² ,1.5,_4+i³ ,1.9, ϕ_i,n = 0.5,_4+i³ ,1.0,_2+i⁵ ,2.0,_3+i⁷ on our method.

All numerical computations were carried out using Matlab version R2019(b). We plot the graphs of errors against the number of iterations in each case. The stopping criterion used for our computation is ∥x_n+1−x_n∥<10⁻⁹.The numerical results are reported in Figures 3.4, 3.5, 3.6 and 3.7 and Tables 3.2.11, 3.2.12, 3.2.13 and 3.2.14. In Tables 3.2.11-3.2.14,

“Iter.” means the number of iterations while “CPU” means the CPU time in seconds.

Example 3.3.11. Let H, H_i =R² for i= 0,1,2, with H =H₀. We define the mappings F = F₀ : R² ×R² → R, F₁ : R² ×R² → R and F₂ : R² ×R² → R respectively by F(x, y) = −3x² +xy+ 2y², F1(x, y) = −4x² +xy+ 3y² and F2(x, y) = −5y² + 2y+ 5xy−5xy² for each x = (x₁, x₂) ∈ R² and y = (y₁, y₂) ∈ R². Also, for i = 0,1,2, let ϕ₀ = ϕ : R² × R² → R, ϕ₁ : R² × R² → R and ϕ₂ : R² × R² → R be defined by ϕ(x, y) = x² −xy, ϕ1(x, y) = 2x(x−y) and ϕ2(x, y) = 5y² −2x respectively for each x = (x₁, x₂) ∈ R² and y = (y₁, y₂) ∈ R². For some r > 0, we obtain by some simple calculation that

v =T_r^F,ϕu= 1

4r+ 1u, y =T_r^F¹^,ϕ¹x= 1

1 + 5rx and w=T_r^F²^,ϕ²z = z−2r 1 + 5r.

Let A_i :R² →R² be given by A_i(x) = _i+1^x where x= (x₁, x₂)∈R². Let S_j :C →C(B) be defined by

S_jx:= −3

2j x, j= 1,2, ...

It is easy to see that S_j is demicontractive for each j = 1,2, . . . .

Table 3.2.11. Numerical results for Example 3.3.11 (Experiment 1).

Cases θ_i,n = _2+i¹ θ_i,n= 1 θ_i,n =

2 3+i

θ_i,n = 1.5

θ_i,n =

3 4+i

1 CPU

time(sec)

0.0155 0.0128 0.0096 0.0140 0.0150

No of Iter. 14 14 14 14 14

2 CPU time

(sec)

0.0136 0.0088 0.0128 0.0139 0.0150

No of Iter. 14 14 14 14 14

3 CPU time

(sec)

0.0137 0.0089 0.0145 0.0162 0.0161

No of Iter. 14 14 14 14 14

4 CPU time

(sec)

0.0135 0.0123 0.0097 0.0146 0.0151

No of Iter. 14 14 14 14 14

Table 3.2.12. Numerical results for Example 3.3.11 (Experiment 2).

Cases ϕi,n = 0.5 ϕi,n =

3 4+i

ϕi,n = 1.0

θi,n =

5 2+i

θi,n = 2.0

1 CPU

time(sec)

0.0138 0.0090 0.0132 0.0156 0.0092

No of Iter. 14 14 14 14 14

2 CPU

time(sec)

0.0139 0.0089 0.0128 0.0144 0.0140

No of Iter. 14 14 14 14 14

3 CPU

time(sec)

0.0139 0.0091 0.0148 0.0151 0.0093

No of Iter. 14 14 14 14 14

4 CPU

time(sec)

0.0136 0.0088 0.0140 0.0152 0.0092

No of Iter. 14 14 14 14 14

The next example is in the framework of an infinite dimensional Hilbert spaces.

Example 3.3.12. Let H, H_i = ℓ₂ for i = 0,1,2, ... be the linear spaces whose elements consists of 2-summable sequences (x₁, x₂, ..., x_i, ...) of scalars, i.e.,

ℓ₂ ={x:x= (x₁, x₂, ..., x_i, ...) and

∞

i=1

|x_i|² <∞}, with an inner product ⟨., .⟩:ℓ₂×ℓ₂ →R defined by

⟨x, y⟩=

∞

i=1

x_iy_i where x={x_i}^∞_i=1, y ={y_i}^∞_i=1 ∈ℓ₂.

For i = 0,1,2, let the mapping A_i : ℓ₂ → ℓ₂ be defined by A_ix = ^x₃¹,^x₃², ..., ^x₃^m, ...

for all x = {x_m}^∞_m=1 ∈ ℓ₂ and A^∗_i : ℓ₂ → ℓ₂ be defined by A^∗_iz = ^z₃¹,^z₃², ...,^z₃^m, ...

for

0 2 4 6 8 10 12 14 Iteration number (n)

10^-10 10^-8 10^-6 10^-4 10^-2 10⁰ 10²

Errors

Alg. 3.1 (_i,n = 1/(2 + i)) Alg. 3.1 (_i,n = 1) Alg. 3.1 (

i,n = 2/(3 + i)) Alg. 3.1 (_i,n = 1.5) Alg. 3.1 (_i,n = 3/(4 + i)) Alg. 3.1 (_i,n = 1.9)

0 2 4 6 8 10 12 14

Iteration number (n) 10^-10

10^-8 10^-6 10^-4 10^-2 10⁰ 10²

Errors

Alg. 3.1 (_i,n = 1/(2 + i)) Alg. 3.1 (_i,n = 1) Alg. 3.1 (

i,n = 2/(3 + i)) Alg. 3.1 (_i,n = 1.5) Alg. 3.1 (_i,n = 3/(4 + i)) Alg. 3.1 (_i,n = 1.9)

0 2 4 6 8 10 12 14

Iteration number (n) 10^-10

10^-8 10^-6 10^-4 10^-2 10⁰ 10²

Errors

Alg. 3.1 (_i,n = 1/(2 + i)) Alg. 3.1 (

i,n = 1) Alg. 3.1 (_i,n = 2/(3 + i)) Alg. 3.1 (_i,n = 1.5) Alg. 3.1 (_i,n = 3/(4 + i)) Alg. 3.1 (_i,n = 1.9)

0 2 4 6 8 10 12 14

Iteration number (n) 10^-10

10^-8 10^-6 10^-4 10^-2 10⁰ 10²

Errors

Alg. 3.1 (_i,n = 1/(2 + i)) Alg. 3.1 (

i,n = 1) Alg. 3.1 (_i,n = 2/(3 + i)) Alg. 3.1 (_i,n = 1.5) Alg. 3.1 (_i,n = 3/(4 + i)) Alg. 3.1 (_i,n = 1.9)

Figure 3.4: Top left: Case 1; Top right: Case 2; Bottom left: Case 3; Bottom right: Case 4.

all z = {z_m}^∞_m=1 ∈ ℓ₂. Define the mapping F_i : ℓ₂ × ℓ₂ → R by F_i,= F such that F(x, y) = −x²+y², ∀ x = {x_i}^∞_i=1, y = {y_i}^∞_i=1. and ϕ_i = 0, for each i = 0,1,2. It is easy to see that

T_r^(F,ϕ)x= 1−r 5r+ 1x.

Also, for j = 1,2, ..., we define S_j :C →CB(ℓ₂) by S_jx=

0, x 5j

∀j = 1,2, ...

It is easy to see that S_j is 0-demicontractive for each j = 1,2, ... and Fix(S_j) ={0}.

Table 3.2.13. Numerical results for Example 3.3.12 (Experiment 1).

0 2 4 6 8 10 12 14 Iteration number (n)

10^-10 10^-8 10^-6 10^-4 10^-2 10⁰ 10²

Errors

Alg. 3.1 (_i,n = 0.5) Alg. 3.1 (_i,n = 3/(4 + i)) Alg. 3.1 (

i,n = 1.0) Alg. 3.1 (_i,n = 5/(2 + i)) Alg. 3.1 (_i,n = 2.0) Alg. 3.1 (_i,n = 7/(3 + i))

0 2 4 6 8 10 12 14

Iteration number (n) 10^-10

10^-8 10^-6 10^-4 10^-2 10⁰ 10²

Errors

Alg. 3.1 (_i,n = 0.5) Alg. 3.1 (_i,n = 3/(4 + i)) Alg. 3.1 (

i,n = 1.0) Alg. 3.1 (_i,n = 5/(2 + i)) Alg. 3.1 (_i,n = 2.0) Alg. 3.1 (_i,n = 7/(3 + i))

0 2 4 6 8 10 12 14

Iteration number (n) 10^-10

10^-8 10^-6 10^-4 10^-2 10⁰ 10²

Errors

Alg. 3.1 (_i,n = 0.5) Alg. 3.1 (

i,n = 3/(4 + i)) Alg. 3.1 (_i,n = 1.0) Alg. 3.1 (_i,n = 5/(2 + i)) Alg. 3.1 (_i,n = 2.0) Alg. 3.1 (_i,n = 7/(3 + i))

0 2 4 6 8 10 12 14

Iteration number (n) 10^-10

10^-8 10^-6 10^-4 10^-2 10⁰ 10²

Errors

Alg. 3.1 (_i,n = 0.5) Alg. 3.1 (

i,n = 3/(4 + i)) Alg. 3.1 (_i,n = 1.0) Alg. 3.1 (_i,n = 5/(2 + i)) Alg. 3.1 (_i,n = 2.0) Alg. 3.1 (_i,n = 7/(3 + i))

Figure 3.5: Top left: Case 1; Top right: Case 2; Bottom left: Case 3; Bottom right: Case 4.

Cases θ_i,n = _2+i¹ θ_i,n= 1 θ_i,n =

2 3+i

θ_i,n = 1.5

θ_i,n =

3 4+i

1 CPU

time(sec)

0.0544 0.0112 0.0144 0.0130 0.0158

No of Iter. 33 33 33 33 33

2 CPU

time(sec)

0.0129 0.0093 0.0133 0.0133 0.0138

No of Iter. 34 33 34 33 34

3 CPU

time(sec)

0.0134 0.0124 0.0096 0.0142 0.0137

No of Iter. 33 33 33 33 33

4 CPU

time(sec)

0.0161 0.0174 0.0164 0.0166 0.0133

No of Iter. 33 33 33 34 34

Table 3.2.14. Numerical results for Example 3.3.12 (Experiment 2).

0 5 10 15 20 25 30 35 Iteration number (n)

10^-10 10^-8 10^-6 10^-4 10^-2 10⁰ 10²

Errors

Alg. 3.1 (_i,n = 1/(2 + i)) Alg. 3.1 (_i,n = 1) Alg. 3.1 (

i,n = 2/(3 + i)) Alg. 3.1 (_i,n = 1.5) Alg. 3.1 (_i,n = 3/(4 + i)) Alg. 3.1 (_i,n = 1.9)

0 5 10 15 20 25 30 35

Iteration number (n) 10^-10

10^-8 10^-6 10^-4 10^-2 10⁰ 10²

Errors

Alg. 3.1 (_i,n = 1/(2 + i)) Alg. 3.1 (_i,n = 1) Alg. 3.1 (

i,n = 2/(3 + i)) Alg. 3.1 (_i,n = 1.5) Alg. 3.1 (_i,n = 3/(4 + i)) Alg. 3.1 (_i,n = 1.9)

0 5 10 15 20 25 30 35

Iteration number (n) 10^-10

10^-8 10^-6 10^-4 10^-2 10⁰ 10²

Errors

Alg. 3.1 (_i,n = 1/(2 + i)) Alg. 3.1 (

i,n = 1) Alg. 3.1 (_i,n = 2/(3 + i)) Alg. 3.1 (_i,n = 1.5) Alg. 3.1 (_i,n = 3/(4 + i)) Alg. 3.1 (_i,n = 1.9)

0 5 10 15 20 25 30 35

Iteration number (n) 10^-10

10^-8 10^-6 10^-4 10^-2 10⁰ 10²

Errors

Alg. 3.1 (_i,n = 1/(2 + i)) Alg. 3.1 (

i,n = 1) Alg. 3.1 (_i,n = 2/(3 + i)) Alg. 3.1 (_i,n = 1.5) Alg. 3.1 (_i,n = 3/(4 + i)) Alg. 3.1 (_i,n = 1.9)

Figure 3.6: Top left: Case 1; Top right: Case 2; Bottom left: Case 3; Bottom right: Case 4.

Cases ϕ_i,n = 0.5 ϕ_i,n =

3 4+i

ϕ_i,n = 1.0

θ_i,n =

5 2+i

θ_i,n = 2.0

1 CPU

time(sec)

0.0132 0.0100 0.0112 0.0116 0.0118

No of Iter. 33 33 33 33 33

2 CPU

time(sec)

0.0154 0.0122 0.0102 0.0146 0.0138

No of Iter. 33 33 33 33 33

3 CPU

time(sec)

0.0141 0.0100 0.0133 0.0141 0.0140

No of Iter. 33 33 33 33 33

4 CPU

time(sec)

0.0161 0.0169 0.0142 0.0169 0.0160

No of Iter. 33 33 33 34 34

We test these examples under the following experiments:

Experiment 1:

In this experiment, we check the behavior of our method by fixing the other parameters and varying θ_i,n. We do this to check the effects of the parameter θ_i,n on our method.

For Example 3.3.11We consider the following cases for the initial value of x₀ : Case 1 : x₀ = (0.78,1.25);

0 5 10 15 20 25 30 35 Iteration number (n)

10^-10 10^-8 10^-6 10^-4 10^-2 10⁰ 10²

Errors

Alg. 3.1 (_i,n = 1/(2 + i)) Alg. 3.1 (_i,n = 1) Alg. 3.1 (

i,n = 2/(3 + i)) Alg. 3.1 (_i,n = 1.5) Alg. 3.1 (_i,n = 3/(4 + i)) Alg. 3.1 (_i,n = 1.9)

0 5 10 15 20 25 30 35

Iteration number (n) 10^-10

10^-8 10^-6 10^-4 10^-2 10⁰ 10²

Errors

Alg. 3.1 (_i,n = 1/(2 + i)) Alg. 3.1 (_i,n = 1) Alg. 3.1 (

i,n = 2/(3 + i)) Alg. 3.1 (_i,n = 1.5) Alg. 3.1 (_i,n = 3/(4 + i)) Alg. 3.1 (_i,n = 1.9)

0 5 10 15 20 25 30 35

Iteration number (n) 10^-10

10^-8 10^-6 10^-4 10^-2 10⁰ 10²

Errors

Alg. 3.1 (_i,n = 1/(2 + i)) Alg. 3.1 (

i,n = 1) Alg. 3.1 (_i,n = 2/(3 + i)) Alg. 3.1 (_i,n = 1.5) Alg. 3.1 (_i,n = 3/(4 + i)) Alg. 3.1 (_i,n = 1.9)

0 5 10 15 20 25 30 35

Iteration number (n) 10^-10

10^-5 10⁰

Errors

Alg. 3.1 (_i,n = 1/(2 + i)) Alg. 3.1 (_i,n = 1) Alg. 3.1 (

i,n = 2/(3 + i)) Alg. 3.1 (

i,n = 1.5) Alg. 3.1 (_i,n = 3/(4 + i)) Alg. 3.1 (_i,n = 1.9)

Figure 3.7: Top left: Case 1; Top right: Case 2; Bottom left: Case 3; Bottom right: Case 4.

Case 2 : x₀ = (3.78,1.25);

Case 3 : x₀ = (4,2);

Case 4 : x₀ = (−1,−5).

Also, we consider θi,n ∈ {_2+i¹ ,1.0,_3+i² ,1.5,_4+i³ ,1.9}, which satisfies Assumption (A3). We use Algorithm 3.3.5 for the experiment and report the numerical results in Table 3.2.11 and Figure 3.4.

For Example 3.3.12We consider the following cases for the initial value of x₀ : Case 1 : x₀ = (2,1,¹₂,· · ·);

Case 2 : x₀ = (4,−2,1,· · ·);

Case 3 : x0 = (−3,³₅,−₂₅³,· · ·);

Case 4 : x₀ = (6,1,¹₆,· · ·).

Also, we consider θ_i,n ∈ {_2+i¹ ,1.0,_3+i² ,1.5,_4+i³ ,1.9}, which satisfies Assumption (A3). We use Algorithm 3.3.5 for the experiment and report the numerical results in Table 3.2.13 and Figure 3.6.

Experiment 2:

In this experiment, we check the behavior of our method by fixing the other parameters and varying ϕ_i,n.We do this to check the effects of the parameter ϕ_i,n on our method.

For Example 3.3.11: We consider the following cases for the initial value of x₀ : Case 1 : x₀ = (0.78,1.25);

Case 2 : x0 = (3.78,1.25);

Case 3 : x₀ = (4,2);

Case 4 : x₀ = (−1,−5).

Also, we consider ϕ_i,n ∈ {0.5,_4+i³ ,1.0,_2+i⁵ ,2.0,_3+i⁷ }, which satisfies Assumption (A4). We use Algorithm 3.3.5 for the experiment and report the numerical results in Table 3.2.12 and Figure 3.5.

For Example 3.3.12: We consider the following cases for the initial value of x₀ : Case 1 : x₀ = (2,1,¹₂,· · ·);

Case 2 : x₀ = (4,−2,1,· · ·);

Case 3 : x₀ = (−3,³₅,−₂₅³,· · ·);

Case 4 : x₀ = (6,1,¹₆,· · ·).

Chapter 4 Split Monotone Variational Inclusion Problems and Fixed Point Problems

4.0.1 Introduction

In this chapter, we present our result on relaxed double inertial Tseng’s extragradient method with self-adaptive step sizes for solving split monotone variational inclusion problem (SMVIP) involving non-Lipschitz operators and fixed point problem of strict pseudo- contractive mappings. Furthermore, results on generalized split feasibility problem over a solution set of monotone variational inclusion problem was studied. For each of these problems, we propose iterative algorithm with self-adaptive step size for approximating the solution and prove strong convergence theorem. We also give applications of our results and illustrate our algorithms with numerical examples.

4.1 Non-Lipschitz split monotone variational inclu-

Dalam dokumen Iterative approximations of certain nonlinear optimization, generalized eqilibrium and fixed point problems in Hilbert and Banach spaces. (Halaman 96-103)