Abstract
Two new conjugate residual algorithms are presented and analyzed in this article. Specifically, the main functions in the system considered are continuous and monotone. The methods are adaptations of the scheme presented by Narushima et al. (SIAM J Optim 21: 212–230, 2011). By employing the famous conjugacy condition of Dai and Liao (Appl Math Optim 43(1): 87–101, 2001), two different search directions are obtained and combined with the projection technique. Apart from being suitable for solving smooth monotone nonlinear problems, the schemes are also ideal for non-smooth nonlinear problems. By employing basic conditions, global convergence of the schemes is established. Report of numerical experiments indicates that the methods are promising.
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Acknowledgements
We would like to extend our gratitude to the anonymous reviewers for their comments and suggestions that helped in improving the work. Also, our gratitude goes to the entire members of the numerical optimization research group, Bayero University, Kano for their encouragements in the course of this work.
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Appendices
Appendix 1
By applying the same approach as for Algorithm 3.1, we obtain that the sequence \(\{x_k\}\) generated by Algorithm 3.2 is bounded.
Lemma 6.1
Given that Assumption 3.3 holds and \(\{x_{k}\}\) is the sequence generated by Algorithm 3.2, then a constant \({\hat{M}}>0\) exists such that
Proof
From (3.33) we analyze two cases: Case(1): \(t\frac{\left\langle F_k,s_{k-1}\right\rangle \left\langle d_{k-1},{\bar{y}}_{k-1}\right\rangle }{\Vert {\bar{y}}_{k-1}\Vert ^2\left\langle F_k,d_{k-1}\right\rangle }>\xi -\psi \frac{\Vert {\bar{y}}_{k-1}\Vert ^2\Vert s_{k-1}\Vert ^2}{\left\langle s_{k-1},{\bar{y}}_{k-1}\right\rangle ^2}\). Then
which from (3.11), (3.34), (4.15) and the Cauchy Schwartz inequality, we have
Similarly, applying (3.11), (3.34), (4.15) and the Cauchy inequality, we can write
So, using (3.35), (6.3), (6.4) and the Cauchy Schwartz inequality, we obtain
Setting \(\left( 1+2\left( \frac{(L+m)}{m}+\frac{t}{m}\right) \right) \kappa =M_1\), we obtain
Case(2): \(t\frac{\left\langle F_k,s_{k-1}\right\rangle \left\langle d_{k-1},{\bar{y}}_{k-1}\right\rangle }{\Vert {\bar{y}}_{k-1}\Vert ^2\left\langle F_k,d_{k-1}\right\rangle }<\xi -\psi \frac{\Vert {\bar{y}}_{k-1}\Vert ^2\Vert s_{k-1}\Vert ^2}{\left\langle s_{k-1},{\bar{y}}_{k-1}\right\rangle ^2}\). Then
So, using (3.11), (3.34), (4.15) and the Cauchy Schwartz inequality, we have
Similarly, applying (3.11), (3.34), (4.15) and the Cauchy Schwartz inequality, we can write
So, using (3.35), (6.8), (6.9) and the Cauchy Schwartz inequality, we obtain
Again setting \(\left( 1+2\left( \frac{(L+m)^2}{m}+\xi \frac{(L+m)^2}{m^2}+\psi \frac{(L+m)^4}{m^4}\right) \right) \kappa =M_2\), we obtain
Therefore, from (6.6) and (6.11), we have
So, setting \({\hat{M}}=\max \{M_1,M_2 \}\), we obtain the result. \(\square\)
Theorem 6.2
Given Assumptions 3.1–3.3 hold and the sequences \(\{x_k\}\) and \(\{{z}_k\}\) are generated by Algorithm 3.2. Then
The proof is established in a similar manner as Algorithm 3.1.
Appendix 2
Appendix 3
See Table 5.
The twelve test problems used for the experiments reported in Tables 1, 2, 3, 4.
Problem 5.1
Strictly Convex Function [83]. \(F_i(x)=e^{x_i}-1, \quad i=1,2,\ldots ,n\).
Problem 5.2
[85].
-
\(F_1(x)=2x_1+\sin {x_1}-1\),
-
\(F_i(x)=2x_i-x_{i-1}+\sin {x_i}-1\),
-
\(F_n(x)=2x_n+\sin {x_n}-1,\quad i=2,3,\ldots ,n-1\).
Problem 5.3
[86].
-
\(F_1(x)=2.5x_{1}+x_{2}-1\),
-
\(F_i(x)=x_{i-1}+2.5x_{i}+x_{i+1}-1, \quad i=2,\ldots ,n-1\),
-
\(F_n(x)=2x_{n-1}+2.5x_{n}-1\).
Problem 5.4
Exponential Function [87].
-
\(F_1(x)=e^{x_1}-1\),
-
\(F_i(x)=e^{x_i}+x_i-1\), \(\quad i=2,\ldots ,n\).
Problem 5.5
Non-smooth Function [42]. \(F_i(x)=2x_i-\sin {|x_i|}\), \(\quad i=1,2, \ldots ,n\).
Problem 5.6
Decretized Chandrasekhar Equation [88].
-
\(F_i(x)=x_i-\left( 1-\frac{c}{2n} \displaystyle \sum _{j=1}^{n} \frac{\mu _ix_j}{\mu _i+\mu _j}\right) ^{-1},\quad i=1,2,\ldots ,n\),
-
with \(c\in [0,1)\) and \(\mu =\frac{ i-0.5}{n},\) for \(1\le i \le n.\) (c is taken as 0.9 in the experiment).
Problem 5.7
[89] The Function F(x) is given by: \(F(x) = A_1x + b_2\),
where \(b_2 = (\sin x_1-1, \ldots ,\sin x_n-1)^T\), and
\(A_1= \left( \begin{matrix} 2&-1&&&\\ 0&2&-1&&\\ &\ddots & \ddots & \ddots &\\ & &\ddots & \ddots & -1\\ &&&0&2 \end{matrix} \right)\).
Problem 5.8
[85] Nonsmooth Function. \(F_i(x)=x_i-\sin |x_i-1|\), \(\quad i=1, \ldots ,n\).
Problem 5.9
Nonsmooth Function [85]. \(F_i(x)=x_i-2\sin |x_i-1|\), \(\quad i=1, \ldots ,n\).
Problem 5.10
Tridiagonal Exponential Function [90].
-
\(F_1(x)= x_{1}-e^{\left( cos{\frac{x_{1}+x_{2}}{n+1}}\right) }\),
-
\(F_i(x)=x_{i}-e^{\left( cos{\frac{x_{i-1}+x_{i}+x_{i+1}}{n+1}}\right) }, \quad i=2,3,\ldots ,n-1\),
-
\(F_n(x)=x_{n}-e^{\left( cos{\frac{x_{n-1}+x_{n}}{n+1}}\right) }\).
Problem 5.11
The Logarithmic Function obtained from [83]. \(F_i(x)=\ln {(x_i+1)}-\frac{x_i}{n}\), \(\quad i=2, \dots n\).
Problem 5.12
[91]. \(F_i(x)=x_{i}-\frac{1}{n}x_{i}^2+\frac{1}{n}\displaystyle \sum _{j=1}^{n}x_{i}+i\), \(\quad i=1,2,\ldots ,n.\)
Remark 6.3
From the selected 12 problems, it can be observed that some have a diagonal Jacobian. Such a separability of the variables turns the outcomes quite independent of the dimensions, as can be observed in the results of Tables 1, 2, 3, 4. Second, although Problem 5.9 is not globally monotone, it seems that local monotonicity is enough to the good behavior of the proposed algorithms, at least with the selected initial points.
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Waziri, M.Y., Ahmed, K. & Halilu, A.S. Adaptive three-term family of conjugate residual methods for system of monotone nonlinear equations. São Paulo J. Math. Sci. 16, 957–996 (2022). https://doi.org/10.1007/s40863-022-00293-0
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DOI: https://doi.org/10.1007/s40863-022-00293-0
Keywords
- Non-smooth functions
- Backtracking line search
- Projection technique
- Conjugacy condition
- Descent condition