A dynamic distributed conjugate gradient method for variational inequality problem over the common fixed-point constraints

Abstract

In this paper, we propose a dynamic distributed conjugate gradient method for solving the strongly monotone variational inequality problem over the intersection of fixed-point sets of firmly nonexpansive operators. The proposed method allows the independent computation of a firmly nonexpansive operator along with the dynamic weight which is updated at each iteration. This strategy aims to speed up the convergence behavior of the algorithm by updating control factors to drive each iterative step. Under some suitable control conditions on corresponding parameters, we show a strong convergence of the iterate to the unique solution of the considered variational inequality problem. We consider the numerical experiments and discuss some observation points by applying the model to solve the image classification problem via the support vector machine learning.

Appendices

Appendix 1: Parameter combinations of DDCGM

We start with the investigation of several parameter combinations which are chosen as in DDCGM by running the algorithm for 100 iterations.

We present the misclassification rate in percentage and the computational runtime (the number in parenthesis) when constructing the classifier for various choices of the parameters λ_k and β_k, when the parameter $\varphi _{k}=\frac {1}{k+1}$, μ = 1, and the parameter γ_k = 0 in Table 1.

Table 1 Misclassification rate in percentage and CPU time for several choices of parameters λ_k ∈ (1,2) and ${\upbeta }_{k}=\frac {\upbeta }{k+1}$ where β ∈ (0,1) when $\varphi _{k}=\frac {1}{k+1}$, μ = 1, and γ_k = 0

We observe from Table 1 that the combination of ${\upbeta }_{k}=\frac {0.9}{k+1}$ with the relaxation parameter λ_k = 1.8 and λ_k = 1.9 and the combination of ${\upbeta }_{k}=\frac {0.8}{k+1}$ with a relaxation parameter λ_k = 1.9 lead to the lowest misclassification rate with 0.8560%. Moreover, under these three combinations, we notice that the combination of ${\upbeta }_{k}=\frac {0.9}{k+1}$ with λ_k = 1.8 yields the fast CPU time of 130.15 sec. It can be seen that, in each choice of λ_k, the amount of CPU time increased as the value of β_k increased.

In Table 2, we present the misclassification rate and CPU time for the combination of several choices of parameters φ_k and λ_k when the the parameters μ = 1 and γ_k = 0, and the parameter ${\upbeta }_{k}=\frac {0.9}{k+1}$, which is the best choice from Table 1. It can be observed that the least misclassification rate and the least CPU time were achieved when the parameter φ_k was quite small. The best classifier performance of 0.8560% misclassification rate was obtained by the choices of λ_k ∈ [1.7, 1.9]. The least computational runtime of 125.58 sec. was obtained from the combination of λ = 1.9 with $\varphi _{k}=\frac {0.2}{k+1}$.

Table 2 Misclassification rate in percentage and CPU time for several choices of parameters λ_k ∈ (1,2) and $\varphi _{k}=\frac {\varphi }{k+1}$ where φ ∈ (0,1) when ${\upbeta }_{k}=\frac {0.9}{k+1}$, μ = 1, and γ_k = 0

Table 3 shows the misclassification rate in percentage and CPU time for several choices of parameters μ and γ_k when the parameters $\lambda _{k}=1.9, \varphi _{k}=\frac {0.2}{k+1}$, and ${\upbeta }_{k}=\frac {0.9}{k+1}$. The best misclassification rate of 0.8056% was obtained by the combination of μ ∈ [1.3, 1.9] with each parameter $\gamma _{k}=\frac {\gamma }{k+1}$ where γ ∈ [0, 0.1]. The least computational runtime was observed for the choice μ = 1.9 with $\gamma _{k}=\frac {0.005}{k+1}$ with 125.98 sec.

Table 3 Misclassification rate in percentage and CPU time for several choices of parameters μ ∈ (0,2) and $\gamma _{k}=\frac {\gamma }{k+1}$ where γ ∈ [0,1) when λ_k = 1.9, $\varphi _{k}=\frac {0.2}{k+1}$, and ${\upbeta }_{k}=\frac {0.3}{k+1}$

Appendix 2: Parameter combinations HSDM [37]

In this section, we present some parameter combinations of the hybrid steepest descent method (HSDM) [37]. We let the operators F and define the operator $T={\sum }_{i=1}^{2m}\omega _{i} T_{i},$ where $\omega _{i}(x)=\frac {1}{2m}$ for all i = 1,…, 2m.

Table 4 shows the misclassification rate in percentage and CPU time for several choices of parameters μ and β_k. The best misclassification rate of 1.2085% and least computational time was obtained by the combination of μ = 1.1 and β_k = 0.9.

Table 4 Misclassification rate in percentage and CPU time for several choices of parameters μ ∈ (0,2) and ${\upbeta }_{k}=\frac {\upbeta }{k+1}$, where β ∈ (0,1)

Appendix 3: Parameter combinations of Iiduka’s [20] HTCGM

In this section, we present some parameter combinations of the hybrid three-term conjugate gradient method (HTCGM) [20, Algorithm 6]. We let the operators F and define the operator $T\! =\! {\sum }_{i=1}^{2m+1}\omega _{i} T_{i},$ where $\omega _{i}(x)=\frac {1}{2m+1}$ for all i = 1,…, 2m + 1.

In Table 5, we present the misclassification rate and CPU time for the combination of several choices of parameters φ_k and β_k when performing HTCGM with the parameters μ = 1 and γ_k = 0. The best classifier performance of 1.1581% misclassification rate was obtained by the choices of $\varphi _{k}=\frac {0.8}{k+1}$ and ${\upbeta }_{k}=\frac {0.8}{k+1}$ with the least CPU time 241.34 sec.

Table 5 Misclassification rate in percentage and CPU time for several choices of parameters $\varphi _{k}=\frac {\varphi }{k+1}$ and ${\upbeta }_{k}=\frac {\upbeta }{k+1}$, where φ,β∈ (0,1], μ = 1, and γ_k = 0

Table 6 shows the misclassification rate in percentage and CPU time for several choices of parameters μ and γ_k when performing HTCGM with the parameters $\varphi _{k}=\frac {0.8}{k+1}$ and ${\upbeta }_{k}=\frac {0.8}{k+1}$. The best misclassification rate of 1.1581% and least computational time were obtained by the combination of μ = 1 and γ_k = 0.01.

Table 6 Misclassification rate in percentage and CPU time for several choices of parameters μ ∈ (0,2) and $\gamma _{k}=\frac {\gamma }{k+1}$ where γ ∈ [0,1) when $\varphi _{k}=\frac {0.8}{k+1}$ and ${\upbeta }_{k}=\frac {0.8}{k+1}$

Appendix 4: Parameter combinations of Cegielski’s [6] ESSPM

In this section, we present some parameter combinations of the extrapolated simultaneous subgradient projection (ESSPM) [6]. All dataset and experimental settings are the same as above.

In Table 7, we present the misclassification rate and CPU time for the combination of several choices of parameters λ_k and β_k when performing ESSPM with the identically constant weight with the parameter μ = 1. The best classifier performance of 1.1581% misclassification rate was obtained by the choices of λ_k = 1.1 and ${\upbeta }_{k}=\frac {0.9}{k+1}$.

Table 7 Misclassification rate in percentage and CPU time for several choices of parameters λ_k ∈ (1,2) and ${\upbeta }_{k}=\frac {\upbeta }{k+1}$ where β ∈ (0,1) when $\varphi _{k}=\frac {1}{k+1}$, μ = 1, and γ_k = 0

Finally, Table 8 shows the misclassification rate in percentage and CPU time for several choices of parameters μ and λ_k when performing ESSPM with the identically constant weight with the parameter ${\upbeta }_{k}=\frac {0.9}{k+1}$. It can be observed that the least misclassification rate was achieved when the parameter μ ≥ 1. The best misclassification rate of 1.1581% and least computational time were obtained by the combination of μ = 1 and λ_k = 1.9.

Table 8 Misclassification rate in percentage and CPU time for several choices of parameters μ ∈ (0,2) and λ_k ∈ (1,2) when ${\upbeta }_{k}=\frac {0.9}{k+1}$