# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020152
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## A self adaptive inertial algorithm for solving split variational inclusion and fixed point problems with applications

 School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

* Corresponding author: Oluwatosin Temitope Mewomo

Received  November 2019 Revised  March 2020 Early access October 2020

We propose a general iterative scheme with inertial term and self-adaptive stepsize for approximating a common solution of Split Variational Inclusion Problem (SVIP) and Fixed Point Problem (FPP) for a quasi-nonexpansive mapping in real Hilbert spaces. We prove that our iterative scheme converges strongly to a common solution of SVIP and FPP for a quasi-nonexpansive mapping, which is also a solution of a certain optimization problem related to a strongly positive bounded linear operator. We apply our proposed algorithm to the problem of finding an equilibrium point with minimal cost of production for a model in industrial electricity production. Numerical results are presented to demonstrate the efficiency of our algorithm in comparison with some other existing algorithms in the literature.

Citation: Timilehin Opeyemi Alakoya, Lateef Olakunle Jolaoso, Oluwatosin Temitope Mewomo. A self adaptive inertial algorithm for solving split variational inclusion and fixed point problems with applications. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020152
##### References:

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##### References:
Example 5.1, Top Left: $N = 50$; Top Left: $N = 100$; Bottom Left: $N = 500$; Bottom Right: $N = 1000$
Example 5.2, Top Left: Case I; Top Left: Case II; Bottom Left: Case III; Bottom Right: Case IV
Example 5.3, Top Left: Choice (i); Top Left: Choice (ii); Bottom Left: Choice (iii); Bottom Right: Choice (iv)
Numerical results for Example 5.1
 No of Iteration CPU time (sec) $N= 50$ 19 0.0289 $N=100$ 19 0.0386 $N=500$ 41 0.1386 $N=1000$ 138 0.3523
 No of Iteration CPU time (sec) $N= 50$ 19 0.0289 $N=100$ 19 0.0386 $N=500$ 41 0.1386 $N=1000$ 138 0.3523
Numerical results for Example 5.2
 Algorithm 3.1 Algorithm 1.1 Algorithm 1.2 Case I CPU time (sec) 0.0021 0.0071 0.0036 $x_0 = 1, x_1 = 0.5$ No of Iter. 8 22 16 Case II CPU time (sec) 0.0021 0.0041 0.0047 $x_0 = -0.5, x_1 = 2$ No. of Iter. 9 25 17 Case III CPU time (sec) 0.0044 0.0532 0.0095 $x_0 = 5, x_1 = 10$ No of Iter. 10 28 19 Case IV CPU time (sec) 0.0062 0.0589 0.0071 $x_0 = -5, x_1 = 2$ No of Iter. 10 27 19
 Algorithm 3.1 Algorithm 1.1 Algorithm 1.2 Case I CPU time (sec) 0.0021 0.0071 0.0036 $x_0 = 1, x_1 = 0.5$ No of Iter. 8 22 16 Case II CPU time (sec) 0.0021 0.0041 0.0047 $x_0 = -0.5, x_1 = 2$ No. of Iter. 9 25 17 Case III CPU time (sec) 0.0044 0.0532 0.0095 $x_0 = 5, x_1 = 10$ No of Iter. 10 28 19 Case IV CPU time (sec) 0.0062 0.0589 0.0071 $x_0 = -5, x_1 = 2$ No of Iter. 10 27 19
Numerical results for Example 5.3
 Algorithm 3.1 Algorithm 1.1 Choice (i) CPU time (sec) 1.7859 5.1231 No of Iter. 11 23 Choice (ii) CPU time (sec) 1.4997 13.3981 No. of Iter. 13 27 Choice (iii) CPU time (sec) 2.6789 9.1093 No of Iter. 7 12 Choice (iv) CPU time (sec) 6.3222 24.5622 No of Iter. 11 24
 Algorithm 3.1 Algorithm 1.1 Choice (i) CPU time (sec) 1.7859 5.1231 No of Iter. 11 23 Choice (ii) CPU time (sec) 1.4997 13.3981 No. of Iter. 13 27 Choice (iii) CPU time (sec) 2.6789 9.1093 No of Iter. 7 12 Choice (iv) CPU time (sec) 6.3222 24.5622 No of Iter. 11 24
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