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# Self adaptive inertial relaxed $CQ$ algorithms for solving split feasibility problem with multiple output sets

The first author is supported by the Petchra Pra Jom Klao Ph.D. Research Scholarship from King Mongkut's University of Technology Thonburi grant No.37/2561

• In this paper, we propose two new self-adaptive inertial relaxed $CQ$ algorithms for solving the split feasibility problem with multiple output sets in the framework of real Hilbert spaces. The proposed algorithms involve computing projections onto half-spaces instead of onto the closed convex sets, and the advantage of the self-adaptive step size introduced in our algorithms is that it does not require the computation of operator norm. We establish and prove weak and strong convergence theorems for the iterative sequences generated by the introduced algorithms for solving the aforementioned problem. Moreover, we apply the new results to solve some other problems. Finally, we present some numerical examples to illustrate the implementation of our algorithms and compared them to some existing results.

Mathematics Subject Classification: 47H09, 65J15, 65K05, 65K10, 49J52.

 Citation: • • Figure 1.  Comparison of Algorithm 1, Algorithm 3, Scheme (16), Scheme (17) and Scheme (5.1) for different choices of $\epsilon$

Figure 2.  Comparison of Algorithm 5, Scheme (13), Scheme (14) and Scheme (17) for different choices of initial points

Table 1.  Algorithm 1 and Algorithm 3 for $\epsilon = 10^{-6}$ and different choices of $\rho_{1}^{n}, \rho_{2}^{n}$ and $\theta$

 $\rho_{1}^{n}=\frac{3n}{4n+1}=\rho_{2}^{n}$ $\rho_{1}^{n}=\frac{n}{2n+1}=\rho_{2}^{n}$ $\rho_{1}^{n}=\frac{n}{2n+1}=\rho_{2}^{n}$ $\rho_{1}^{n}=\frac{3n}{20n+1}=\rho_{2}^{n}$ Iter.(n) CPU(s) En Iter.(n) CPU(s) En Iter.(n) CPU(s) En Iter.(n) CPU(s) En Algorithm 1 22 0.018314 8.41E-07 32 0.025566 7.17E-07 53 0.024086 9.40E-07 74 0.02651 9.69E-07 Algorithm 3 83 0.023672 9.67E-07 91 0.042918 9.61E-07 157 0.028161 9.88E-07 207 0.034091 9.80E-07 $\theta=0$ $\theta=0.15$ $\theta=0.25$ $\theta=0.5$ Iter.(n) CPU(s) En Iter.(n) CPU(s) En Iter.(n) CPU(s) En Iter.(n) CPU(s) En Algorithm 1 45 0.025492 8.00E-07 57 0.02797 7.78E-07 43 0.029967 9.76E-07 51 0.02619 8.02E-07 Algorithm 3 111 0.026377 9.79E-07 136 0.030002 9.82E-07 91 0.026963 9.81E-07 84 0.024384 9.82E-07

Table 2.  Algorithm 1, Algorithm 3, Scheme (16), Scheme (17) and Scheme (5.1) for different choices of $\epsilon$

 Algorithm 1 Algorithm 3 Scheme (16) Scheme (17) Scheme (5.1) $\epsilon=10^{-6}$ Iter.(n) 24 75 180 111 75 CPU(s) 0.01667 0.02255 0.023436 0.037266 0.029002 $E_n$ 8.25E-07 9.67E-07 9.74E-07 9.82E-07 9.92E-07 $\epsilon=10^{-7}$ Iter.(n) 30 134 174 282 211 CPU(s) 0.01962 0.025028 0.025425 0.026567 0.033771 $E_n$ 6.17E-08 9.83E-08 9.80E-08 9.96E-08 9.99E-08 $\epsilon=10^{-8}$ Iter.(n) 41 276 470 537 770 CPU(s) 0.024448 0.029783 0.033593 0.035215 0.038546 $E_n$ 8.68E-09 9.95E-09 9.84E-09 9.99E-09 9.98E-09 $\epsilon=10^{-9}$ Iter.(n) 49 479 496 2024 2263 CPU(s) 0.026591 0.037697 0.036028 0.039359 0.088713 $E_n$ 6.98E-10 9.93E-10 9.83E-10 1.00E-09 1.00E-09

Table 3.  Comparison of Algorithm 5, Scheme (13), Scheme (14) and Scheme (17)

 Algorithm 5 Scheme (14) Scheme (13) Scheme (17) Iter.(n) $E_n$ CPU(s) Iter.(n) $E_n$ CPU(s) Iter.(n) $E_n$ CPU(s) Iter.(n) $E_n$ CPU(s) 1 1149.360361 1 231.8966545 1 199.2220601 1 640.4875017 2 28.03259245 2 91.40575598 2 69.5163167 2 12.35158962 3 0.811105412 3 32.14832803 3 20.87177641 3 2.434404763 4 0.202894757 4 10.25120406 4 6.548286817 4 0.513146017 5 0.050765569 5 3.137538418 5 2.301982084 5 0.140142486 6 0.012707645 6 1.056033824 6 0.966763557 6 0.083619119 7 0.003183583 7 0.449136255 7 0.488009333 7 0.059127765 8 0.000798736 8 0.228594093 8 0.279757135 8 0.041809171 9 0.000200925 9 0.11848334 9 0.170839653 9 0.029562999 10 5.07839E-05 69.73042 10 0.056872251 10 0.106846934 10 0.020903734 11 0.024488093 11 0.06722031 11 0.014780829 12 0.009391695 12 0.042254471 12 0.010451389 13 0.00322014 13 0.026486762 13 0.007390099 14 0.001009238 14 0.01655399 14 0.005225502 15 0.000311098 15 0.010319917 15 0.003694944 16 0.000109788 16 0.006420625 16 0.002612704 17 4.93538E-05 140.9432 17 0.003988518 17 0.001847463 18 0.002474828 18 0.001306366 19 0.001534285 19 0.000923758 20 0.000950584 20 0.000653215 21 0.000588668 21 0.000461913 22 0.000364417 22 0.00032664 23 0.000225534 23 0.000230986 24 0.000139554 24 0.000163347 25 8.6339E-05 173.0115 25 0.000115517 26 8.16939E-05 132.0709
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