Analysis of a variable metric block coordinate method under proximal errors

Abstract

This paper focuses on an inexact block coordinate method designed for nonsmooth optimization, where each block-subproblem is solved by performing a bounded number of steps of a variable metric proximal–gradient method with linesearch. We improve on the existing analysis for this algorithm in the nonconvex setting, showing that the iterates converge to a stationary point of the objective function even when the proximal operator is computed inexactly, according to an implementable inexactness condition. The result is obtained by introducing an appropriate surrogate function that takes into account the inexact evaluation of the proximal operator, and assuming that such function satisfies the Kurdyka–Łojasiewicz inequality. The proof technique employed here may be applied to other new or existing block coordinate methods suited for the same class of optimization problems.

Appendices

Appendix A Proof of Lemma 8

Proof of Lemma 8

Applying the descent lemma [36, Proposition A.24] yields

$$\begin{aligned} f_0(\hat{u}^{(k)})\ge f_0({\tilde{u}}^{(k)}) - \nabla f_0(\hat{u}^{(k)})^T({\tilde{u}}^{(k)}-\hat{u}^{(k)}) -\frac{M}{2} \Vert {\tilde{u}}^{(k)}-\hat{u}^{(k)}\Vert ^2. \end{aligned}$$

(A1)

By setting $z_i^{(k,\bar{\ell }_i^{(k)})}=x_i^{(k,\bar{\ell }_i^{(k)})}-\alpha _i^{(k,\bar{\ell }_i^{(k)})} (D_i^{(k,\bar{\ell }_i^{(k)})})^{-1}\nabla _i f_0({\tilde{x}}_i^{(k,\bar{\ell }_i^{(k)})})$, $\epsilon _i^{(k)}=\epsilon _i^{(k,\bar{\ell }_i^{(k)})}$, and applying some calculus rules for the $\epsilon _i^{(k)}-$subdifferential (see [33, Appendix A]), the inexactness condition (20) implies the existence of a vector $e_i^{(k)}\in {\mathbb {R}}^{n_i}$ such that

$$\begin{aligned} -\frac{1}{\alpha _i^{(k,\bar{\ell }_i^{(k)})}} D_i^{(k,\bar{\ell }_i^{(k)})}\left( u_i^{(k,\bar{\ell }_i^{(k)})}-z_i^{(k,\bar{\ell }_i^{(k)})} + e_i^{(k)}\right)&\in \partial _{\epsilon _i^{(k)}}f_i(u_i^{(k,\bar{\ell }_i^{(k)})}) \end{aligned}$$

(A2)

$$\begin{aligned} \frac{1}{2\alpha _i^{(k,\bar{\ell }_i^{(k)})}} \Vert e_i^{(k)}\Vert _{D_i^{(k,\bar{\ell }_i^{(k)})}}^2&\le \epsilon _i^{(k)}. \end{aligned}$$

(A3)

By employing Definition 4 with ${{\mathcal {F}}}=f_i$, $u = \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}$, $z = u_i^{(k,\bar{\ell }_i^{(k)})}$, $\epsilon =\epsilon _i^{(k)}$, and w given by (A2), we have

$$\begin{aligned} f_i(\hat{u}_i^{(k,\bar{\ell }_i^{(k)})})&\ge f_i(u_i^{(k,\bar{\ell }_i^{(k)})})\\&-\frac{1}{\alpha _i^{(k,\bar{\ell }_i^{(k)})}} \left( u_i^{(k,\bar{\ell }_i^{(k)})}-x_i^{(k,\bar{\ell }_i^{(k)})}\right) ^TD_i^{(k,\bar{\ell }_i^{(k)})}\left( \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\right) \\&-\nabla _i f_0({\tilde{x}}_i^{(k,\bar{\ell }_i^{(k)})})^T\left( \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\right) \\&-\frac{1}{\alpha _i^{(k,\bar{\ell }_i^{(k)})}}(e_i^{(k)})^TD_i^{(k,\bar{\ell }_i^{(k)})}\left( \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\right) -\epsilon _i^{(k)}. \end{aligned}$$

Summing the previous inequality over $i=1,\ldots ,p$ with (A1) and recalling the definition of the points $\hat{u}^{(k)}$, ${\tilde{u}}^{(k)}$ leads to

$$\begin{aligned}&f(\hat{u}^{(k)})\ge f({\tilde{u}}^{(k)})- \nabla f_0(\hat{u}^{(k)})^T({\tilde{u}}^{(k)}-\hat{u}^{(k)})-\sum _{i=1}^{p}\nabla _i f_0({\tilde{x}}_i^{(k,\bar{\ell }_i^{(k)})})^T\left( \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\right) \\&-\sum _{i=1}^{p}\frac{1}{\alpha _i^{(k,\bar{\ell }_i^{(k)})}} \left( u_i^{(k,\bar{\ell }_i^{(k)})}-x_i^{(k,\bar{\ell }_i^{(k)})}\right) ^TD_i^{(k,\bar{\ell }_i^{(k)})}\left( \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\right) \\&-\sum _{i=1}^{p}\frac{1}{\alpha _i^{(k,\bar{\ell }_i^{(k)})}}(e_i^{(k)})^TD_i^{(k,\bar{\ell }_i^{(k)})}\left( \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\right) -\frac{M}{2} \Vert {\tilde{u}}^{(k)}-\hat{u}^{(k)}\Vert ^2-\sum _{i=1}^{p}\epsilon _i^{(k)}. \end{aligned}$$

By rearranging terms and applying the Cauchy-Schwarz inequality, we get

$$\begin{aligned}&f(\hat{u}^{(k)})\ge f({\tilde{u}}^{(k)})-\sum _{i=1}^{p}\Vert \nabla _i f_0({\tilde{x}}_i^{(k,\bar{\ell }_i^{(k)})})-\nabla _if_0(\hat{u}^{(k)})\Vert \Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\Vert \nonumber \\&-\sum _{i=1}^{p}\frac{1}{\alpha _i^{(k,\bar{\ell }_i^{(k)})}} \left( u_i^{(k,\bar{\ell }_i^{(k)})}-x_i^{(k,\bar{\ell }_i^{(k)})}\right) ^TD_i^{(k,\bar{\ell }_i^{(k)})}\left( \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\right) \nonumber \\&-\sum _{i=1}^{p}\frac{1}{\alpha _i^{(k,\bar{\ell }_i^{(k)})}}(e_i^{(k)})^TD_i^{(k,\bar{\ell }_i^{(k)})}\left( \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\right) -\frac{M}{2} \Vert {\tilde{u}}^{(k)}-\hat{u}^{(k)}\Vert ^2-\sum _{i=1}^{p}\epsilon _i^{(k)}. \end{aligned}$$

(A4)

At this point, for all $i=1,\ldots ,p$, we can write the following chain of inequalities:

$$\begin{aligned}&\Vert \nabla _i f_0({\tilde{x}}_i^{(k,\bar{\ell }_i^{(k)})})-\nabla _if_0(\hat{u}^{(k)})\Vert \Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\Vert \\&\le {M}\Vert {\tilde{x}}_i^{(k,\bar{\ell }_i^{(k)})}-\hat{u}^{(k)}\Vert \Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\Vert \\&\le {M} \left( \sum _{j=1}^{i-1}\Vert x_j^{(k,L_j^{(k)})}-x_j^{(k,\bar{\ell }_j^{(k)})}\Vert +\Vert x_j^{(k,\bar{\ell }_j^{(k)})}-\hat{u}_j^{(k,\bar{\ell }_j^{(k)})}\Vert \right) \Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\Vert \\&\quad +{M}\left( \sum _{j=i+1}^{p}\Vert x_j^{(k)}-x_j^{(k,\bar{\ell }_j^{(k)})}\Vert +\Vert x_j^{(k,\bar{\ell }_j^{(k)})}-\hat{u}_j^{(k,\bar{\ell }_j^{(k)})}\Vert \right) \Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\Vert \\&\quad +{M}\Vert x_i^{(k,\bar{\ell }_i^{(k)})}-\hat{u}_i^{(k,\bar{\ell }_i^{(k)})}\Vert \Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\Vert , \end{aligned}$$

where the first inequality follows by means of Assumption 1(iii) and the second one is obtained by applying the definition of points ${\tilde{x}}_i^{(k,\bar{\ell }_i^{(k)})},\hat{u}^{(k)}$, the basic inequality $\sqrt{a+b}\le \sqrt{a}+\sqrt{b}$, and the triangular inequality. We can further proceed as follows

$$\begin{aligned}&\Vert \nabla _i f_0({\tilde{x}}_i^{(k,\bar{\ell }_i^{(k)})})-\nabla _if_0(\hat{u}^{(k)})\Vert \Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\Vert \nonumber \\&\le {M} \left( \sum _{j=1}^{i-1}\sum _{\ell =\bar{\ell }_j^{(k)}}^{L_j^{(k)}-1}\Vert x_j^{(k,\ell +1)}-x_j^{(k,\ell )}\Vert +\sum _{j=1}^{i-1}\Vert x_j^{(k,\bar{\ell }_j^{(k)})}-\hat{u}_j^{(k,\bar{\ell }_j^{(k)})}\Vert \right) \Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\Vert \nonumber \\&\quad +{M}\left( \sum _{j=i+1}^{p}\sum _{\ell =0}^{\bar{\ell }_j^{(k)}-1}\Vert x_j^{(k,\ell +1)}-x_j^{(k,\ell )}\Vert +\sum _{j=i+1}^{p}\Vert x_j^{(k,\bar{\ell }_j^{(k)})}-\hat{u}_j^{(k,\bar{\ell }_j^{(k)})}\Vert \right) \Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\Vert \nonumber \\&\quad +{M}\Vert x_i^{(k,\bar{\ell }_i^{(k)})}-\hat{u}_i^{(k,\bar{\ell }_i^{(k)})}\Vert \Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\Vert \nonumber \\&\le {M}\Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\Vert \sum _{j=1}^{p}\Vert x_j^{(k,\bar{\ell }_j^{(k)})}-\hat{u}_j^{(k,\bar{\ell }_j^{(k)})}\Vert \nonumber \\&\quad +2{M}\Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\Vert \sum _{j=1}^{p}\sum _{\ell =0}^{L_j^{(k)}-1}\Vert x_j^{(k,\ell +1)}-x_j^{(k,\ell )}\Vert \end{aligned}$$

(A5)

where again we have used the triangular inequality to get the first inequality. Summing (A5) over $i=1,\ldots ,p$ leads to

$$\begin{aligned}&\sum _{i=1}^{p}\Vert \nabla _i f_0({\tilde{x}}_i^{(k,\bar{\ell }_i^{(k)})})-\nabla _if_0(\hat{u}^{(k)})\Vert \Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\Vert \\&\le {M}\sum _{i=1}^{p}\Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\Vert \sum _{j=1}^{p}\Vert x_j^{(k,\bar{\ell }_j^{(k)})}-\hat{u}_j^{(k,\bar{\ell }_j^{(k)})}\Vert \\&+2{M}\sum _{i=1}^{p}\Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\Vert \sum _{j=1}^{p}\sum _{\ell =0}^{L_j^{(k)}-1}\Vert x_j^{(k,\ell +1)}-x_j^{(k,\ell )}\Vert . \end{aligned}$$

We can now employ inequalities (22)–(23)–(25) to deduce the following upper bound

$$\begin{aligned}&\sum _{i=1}^{p}\Vert \nabla _i f_0({\tilde{x}}_i^{(k,\bar{\ell }_i^{(k)})})-\nabla _if_0(\hat{u}^{(k)})\Vert \Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\Vert \\&\le {M}\alpha _{\max }\mu \sqrt{2\tau \left( 1+\frac{\tau }{2}\right) }\left( \sum _{i=1}^{p}\sqrt{-h_i^{(k,\bar{\ell }_i^{(k)})}(u_i^{(k,\bar{\ell }_i^{(k)})})}\right) ^2\\&\quad +4{M}\alpha _{\max }\mu \sqrt{\tau (1+\tau )}\left( \sum _{i=1}^{p}\sum _{\ell =0}^{L_i^{(k)}-1}\sqrt{-h_i^{(k,\ell )}(u_i^{(k,\ell )})}\right) ^2\\&\le {\tilde{c}}_1\left( \sum _{i=1}^{p}\sum _{\ell =0}^{L_i^{(k)}-1}\sqrt{-h_i^{(k,\ell )}(u_i^{(k,\ell )})}\right) ^2, \end{aligned}$$

where ${\tilde{c}}_1=L\alpha _{\max }\mu \left( 4\sqrt{\tau (1+\tau )}+\sqrt{2\tau \left( 1+\frac{\tau }{2}\right) }\right) \ge 0$. Multiplying and dividing the quantity inside the round brackets by $\sum _{i=1}^{p}L_i^{(k)}$, applying the Jensen inequality to the function $\phi (t)=t^2$ and recalling that $L_i^{(k)}\le L_i$ for all $k\in {\mathbb {N}}$, $i=1,\ldots ,p$, allows to obtain

$$\begin{aligned} \sum _{i=1}^{p}\Vert \nabla _i f_0({\tilde{x}}_i^{(k,\bar{\ell }_i^{(k)})}){} & {} -\nabla _if_0(\hat{u}^{(k)})\Vert \Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\Vert \nonumber \\ {}{} & {} \le c_1\sum _{i=1}^{p}\sum _{\ell =0}^{L_i^{(k)}-1}\left( -h_i^{(k,\ell )}(u_i^{(k,\ell )})\right) . \end{aligned}$$

(A6)

where $c_1={\tilde{c}}_1\sum _{i=1}^{p}L_i\ge 0$. Similarly, we can provide the following upper bound

$$\begin{aligned}&\sum _{i=1}^{p}\frac{1}{\alpha _i^{(k,\bar{\ell }_i^{(k)})}} \left( u_i^{(k,\bar{\ell }_i^{(k)})}-x_i^{(k,\bar{\ell }_i^{(k)})}\right) ^TD_i^{(k,\bar{\ell }_i^{(k)})}\left( \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\right) \nonumber \\&\le \frac{\mu }{\alpha _{\min }}\sum _{i=1}^{p}\Vert u_i^{(k,\bar{\ell }_i^{(k)})}-x_i^{(k,\bar{\ell }_i^{(k)})}\Vert \Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\Vert \nonumber \\&\le c_2\sum _{i=1}^{p}\left( -h_i^{(k,\bar{\ell }_i^{(k)})}(u_i^{(k,\bar{\ell }_i^{(k)})})\right) \end{aligned}$$

(A7)

where the first inequality follows again from the Cauchy-Schwarz inequality together with $\alpha _i^{(k,\bar{\ell }_i^{(k)})}\ge \alpha _{\min }$ and $D_i^{(k,\bar{\ell }_i^{(k)})}\in {\mathcal {M}}_\mu$, the second inequality is due to the application of (23)–(24) and $c_2=\frac{2\mu ^2\alpha _{\max }}{\alpha _{\min }}\sqrt{\tau (1+\tau )}\ge 0$. Moreover, from relation (A3), we deduce

$$\begin{aligned} \Vert e_i^{(k)}\Vert ^2 \le \mu \Vert e_i^{(k)}\Vert _{D_i^{(k,\bar{\ell }_i^{(k)})}}^2 \le 2\alpha _i^{(k,\bar{\ell }_i^{(k)})}\mu \epsilon _i^{(k)}\le -\alpha _{\max }\mu \tau h_i^{(k,\bar{\ell }_i^{(k)})}(u_i^{(k,\bar{\ell }_i^{(k)})}). \end{aligned}$$

Then we have

$$\begin{aligned}&\sum _{i=1}^{p}\frac{1}{\alpha _i^{(k,\bar{\ell }_i^{(k)})}}(e_i^{(k)})^TD_i^{(k,\bar{\ell }_i^{(k)})}\left( \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\right) \nonumber \\&\le \frac{\mu }{\alpha _{\min }}\sum _{i=1}^{p}\Vert e_i^{(k)}\Vert \Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-u_i^{(k,\bar{\ell }_i^{(k)})}\Vert \le c_3 \sum _{i=1}^{p}\left( -h_i^{(k,\bar{\ell }_i^{(k)})}(u_i^{(k,\bar{\ell }_i^{(k)})})\right) \end{aligned}$$

(A8)

where the second inequality follows from (23) and (A8), whereas $c_3=\frac{\mu ^2\alpha _{\max }\tau }{\alpha _{\min }}\ge 0$.

Finally, plugging (A6)–(A7)–(A8) inside (A4) gives

$$\begin{aligned} f(\hat{u}^{(k)})&\ge f({\tilde{u}}^{(k)})-c_1\sum _{i=1}^{p}\sum _{\ell =0}^{L_i^{(k)}-1}\left( -h_i^{(k,\ell )}(u_i^{(k,\ell )})\right) -c_2\sum _{i=1}^{p}\left( -h_i^{(k,\bar{\ell }_i^{(k)})}(u_i^{(k,\bar{\ell }_i^{(k)})})\right) \\&\quad -c_3 \sum _{i=1}^{p}\left( -h_i^{(k,\bar{\ell }_i^{(k)})}(u_i^{(k,\bar{\ell }_i^{(k)})})\right) -\frac{{M}\alpha _{\max }\mu \tau }{2}\sum _{i=1}^{p}\left( -h_i^{(k,\bar{\ell }_i^{(k)})}(u_i^{(k,\bar{\ell }_i^{(k)})})\right) \\&\quad -\frac{\tau }{2}\sum _{i=1}^{p}\left( -h_i^{(k,\bar{\ell }_i^{(k)})}(u_i^{(k,\bar{\ell }_i^{(k)})})\right) . \end{aligned}$$

Then the thesis follows by setting $c=c_1+c_2+c_3+\frac{{M}\alpha _{\max }\mu \tau }{2}+\frac{\tau }{2}\ge 0$. $\square$

Appendix B Proof of Lemma 9

Proof of Lemma 9

Using the descent lemma, we obtain

$$\begin{aligned} f_0(\hat{u}^{(k)})\le f_0( x^{(k)})+\nabla f_0( x^{(k)})^T(\hat{u}^{(k)}- x^{(k)})+\frac{{M}}{2}\Vert \hat{u}^{(k)}- x^{(k)}\Vert ^2. \end{aligned}$$

We sum $\sum _{i=1}^{p}f_i(\hat{u}_i^{(k,\bar{\ell }_i^{(k)})})$ to both sides and rearrange terms as follows:

$$\begin{aligned} f(\hat{u}^{(k)})&= f( x^{(k)})+\sum _{i=1}^{p}\left( f_i(\hat{u}_i^{(k,\bar{\ell }_i^{(k)})})-f_i(x_i^{(k)})+\nabla _if_0( x^{(k)})^T(\hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-x_i^{(k)})\right) \\&\quad +\frac{{M}}{2}\Vert \hat{u}^{(k)}- x^{(k)}\Vert ^2\\&= f( x^{(k)})+\sum _{i=1}^{p}\left( f_i(\hat{u}_i^{(k,\bar{\ell }_i^{(k)})})-f_i(x_i^{(k)})+\nabla _if_0({\tilde{x}}_i^{(k,\bar{\ell }_i^{(k)})})^T(\hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-x_i^{(k)})\right) \\&\quad +\sum _{i=1}^{p}(\nabla _if_0(x^{(k)})-\nabla _if_0({\tilde{x}}_i^{(k,\bar{\ell }_i^{(k)})}))^T(\hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-x_i^{(k)})+\frac{{M}}{2}\Vert \hat{u}^{(k)}- x^{(k)}\Vert ^2\\&= f( x^{(k)})+\sum _{i=1}^{p}\left( f_i(\hat{u}_i^{(k,\bar{\ell }_i^{(k)})})-f_i(x_i^{(k,\bar{\ell }_i^{(k)})})+\nabla _if_0({\tilde{x}}_i^{(k,\bar{\ell }_i^{(k)})})^T(\hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-x_i^{(k,\bar{\ell }_i^{(k)})})\right) \\&\quad +\sum _{i=1}^{p}\left( f_i(x_i^{(k,\bar{\ell }_i^{(k)})})-f_i(x_i^{(k)})+\nabla _if_0({\tilde{x}}_i^{(k,\bar{\ell }_i^{(k)})})^T(x_i^{(k,\bar{\ell }_i^{(k)})}-x_i^{{(k)}})\right) \\&\quad +\sum _{i=1}^{p}(\nabla _if_0(x^{(k)})-\nabla _if_0({\tilde{x}}_i^{(k,\bar{\ell }_i^{(k)})}))^T(\hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-x_i^{(k)})+\frac{{M}}{2}\Vert \hat{u}^{(k)}- x^{(k)}\Vert ^2. \end{aligned}$$

By adding the quantity $\sum _{i=1}^{p}(2\alpha _i^{(k,\bar{\ell }_i^{(k)})})^{-1}\Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-x_i^{(k,\bar{\ell }_i^{(k)})}\Vert _{D_i^{(k,\bar{\ell }_i^{(k)})}}^2$, and summing and subtracting the term $\sum _{i=1}^{p}(2\alpha _i^{(k,\bar{\ell }_i^{(k)})})^{-1}\Vert x_i^{(k)}-x_i^{(k,\bar{\ell }_i^{(k)})}\Vert _{D_i^{(k,\bar{\ell }_i^{(k)})}}^2$ to the right-hand side of the previous inequality, we directly get

$$\begin{aligned} f(\hat{u}^{(k)})&\le f( x^{(k)})+\sum _{i=1}^{p}h_i^{(k,\bar{\ell }_i^{(k)})}(\hat{u}_i^{(k,\bar{\ell }_i^{(k)})})+\sum _{i=1}^{p}\left( -h_i^{(k,\bar{\ell }_i^{(k)})}(x_i^{(k)})\right) \nonumber \\&\quad +\sum _{i=1}^{p}(\nabla _if_0(x^{(k)})-\nabla _if_0({\tilde{x}}_i^{(k,\bar{\ell }_i^{(k)})}))^T(\hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-x_i^{(k)})\nonumber \\&\quad +\sum _{i=1}^{p}\frac{1}{2\alpha _i^{(k,\bar{\ell }_i^{(k)})}}\Vert x_i^{(k)}-x_i^{(k,\bar{\ell }_i^{(k)})}\Vert _{D_i^{(k,\bar{\ell }_i^{(k)})}}^2+\frac{{M}}{2}\Vert \hat{u}^{(k)}- x^{(k)}\Vert ^2. \end{aligned}$$

(B9)

By setting $u=x_i^{(k,\bar{\ell }_i^{(k)})}$ and $u=x_i^{(k)}$ in (15), respectively, we obtain

$$\begin{aligned} h_i^{(k,\bar{\ell }_i^{(k)})}(\hat{u}_i^{(k,\bar{\ell }_i^{(k)})})\le 0, \quad -h_i ^{(k,\bar{\ell }_i^{(k)})}(x_i^{(k)})\le -h_i^{(k,\bar{\ell }_i^{(k)})}(\hat{u}_i^{(k,\bar{\ell }_i^{(k)})}). \end{aligned}$$

(B10)

Plugging condition (18) inside the previous inequality leads to

$$\begin{aligned} h_i^{(k,\bar{\ell }_i^{(k)})}(\hat{u}_i^{(k,\bar{\ell }_i^{(k)})})\le 0, \quad -h_i ^{(k,\bar{\ell }_i^{(k)})}(x_i^{(k)})\le -\left( 1+\frac{\tau }{2}\right) h_i^{(k,\bar{\ell }_i^{(k)})}(u_i^{(k,\bar{\ell }_i^{(k)})}). \end{aligned}$$

(B11)

By inserting (B11) inside (B9), we can continue as follows

$$\begin{aligned} f(\hat{u}^{(k)})&\le f( x^{(k)})+\left( 1+\frac{\tau }{2}\right) \sum _{i=1}^{p}\left( -h_i^{(k,\bar{\ell }_i^{(k)})}(u_i^{(k,\bar{\ell }_i^{(k)})})\right) \\&\quad +\sum _{i=1}^{p}\left( \nabla _if_0(x^{(k)})-\nabla _if_0({\tilde{x}}_i^{(k,\bar{\ell }_i^{(k)})})\right) ^T \left( \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-x_i^{(k)}\right) \\&\quad +\sum _{i=1}^{p}\frac{\mu }{2\alpha _{\min }}\Vert x_i^{(k)}-x_i^{(k,\bar{\ell }_i^{(k)})}\Vert ^2+\frac{{M}}{2}\Vert \hat{u}^{(k)}- x^{(k)}\Vert ^2. \end{aligned}$$

A direct application of the triangular inequality yields

$$\begin{aligned} f(\hat{u}^{(k)})&\le f( x^{(k)})+\left( 1+\frac{\tau }{2}\right) \sum _{i=1}^{p}\left( -h_i^{(k,\bar{\ell }_i^{(k)})}(u_i^{(k,\bar{\ell }_i^{(k)})})\right) \nonumber \\&\quad +\sum _{i=1}^{p}\left( \nabla _if_0(x^{(k)})-\nabla _if_0({\tilde{x}}_i^{(k,\bar{\ell }_i^{(k)})})\right) ^T \left( \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-x_i^{(k)}\right) \nonumber \\&\quad +\frac{\mu }{2\alpha _{\min }}\sum _{i=1}^{p}\left( \sum _{\ell =0}^{\bar{\ell }_i^{(k)}-1}\Vert x_i^{(k,\ell +1)}-x_i^{{(k,\ell )}}\Vert \right) ^2\nonumber \\&\quad +\frac{{M}}{2}\sum _{i=1}^{p}\left( \sum _{\ell =0}^{\bar{\ell }_i^{(k)}-1}\Vert x_i^{(\ell +1)}-x_i^{(k,\ell )}\Vert +\Vert x_i^{(k,\bar{\ell }_i^{(k)})}-\hat{u}_i^{(k,\bar{\ell }_i^{(k)})}\Vert \right) ^2. \end{aligned}$$

(B12)

We now provide an upper bound for the third addend in the above inequality, proceeding as follows

$$\begin{aligned}&\left( \nabla _if_0(x^{(k)})-\nabla _if_0({\tilde{x}}_i^{(k,\bar{\ell }_i^{(k)})})\right) ^T\left( \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-x_i^{(k)}\right) \nonumber \\&\le \Vert \nabla _if_0(x^{(k)})-\nabla _if_0({\tilde{x}}_i^{(k,\bar{\ell }_i^{(k)})})\Vert \Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-x_i^{(k)}\Vert \nonumber \\&\le {M}\Vert x^{(k)}-{\tilde{x}}_i^{(k,\bar{\ell }_i^{(k)})}\Vert \Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-x_i^{(k)}\Vert \nonumber \\&\le {M}\Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-x_i^{(k)}\Vert {\left( \sum _{j=1}^{i-1}\Vert x_j^{(k)}-x_j^{(k,L_j^{(k)})}\Vert +\Vert x_i^{(k)}-x_i^{(k,\bar{\ell }_i^{(k)})}\Vert \right) }\nonumber \\&\le {M}\Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-x_i^{(k)}\Vert {\left( \sum _{j=1}^{i-1}\sum _{\ell = 0}^{L_j^{(k)}-1}\Vert x_j^{(k,\ell +1)}-x_j^{(k,\ell )}\Vert +\sum _{\ell =0}^{\bar{\ell }_i^{(k)}-1}\Vert x_i^{(k,\ell +1)}-x_i^{(k,\ell )}\Vert \right) }\nonumber \\&{\le {M}\Vert \hat{u}_i^{(k,\bar{\ell }_i^{(k)})}-x_i^{(k)}\Vert \sum _{j=1}^{i}\sum _{\ell = 0}^{L_j^{(k)}-1}\Vert x_j^{(k,\ell +1)}-x_j^{(k,\ell )}\Vert }\nonumber \\&\le {M}\left( \sum _{\ell =0}^{\bar{\ell }_i^{(k)}-1}\Vert x_i^{(k,\ell +1)}-x_i^{(k,\ell )}\Vert +\Vert x_i^{(k,\bar{\ell }_i^{(k)})}-\hat{u}_i^{(k,\bar{\ell }_i^{(k)})}\Vert \right) \sum _{j=1}^{p}\sum _{\ell =0}^{{L_j^{(k)}-1}}\Vert x_j^{(k,\ell +1)}-x_j^{(k,\ell )}\Vert \end{aligned}$$

(B13)

where the third inequality follows from the definition of ${\tilde{x}}_i^{(k,\bar{\ell }_i^{(k)})}$ in (12), whereas we have used the triangular inequality, $\bar{\ell }_i^{(k)}\le L_i^{(k)}$, and $i\le p$, to get the last three inequalities. If we sum (B13) over $i=1,\ldots ,p$ and apply it to (B12), we come to

$$\begin{aligned} f(\hat{u}^{(k)})&\le f( x^{(k)})+\left( 1+\frac{\tau }{2}\right) \sum _{i=1}^{p}\left( -h_i^{(k,\bar{\ell }_i^{(k)})}(u_i^{(k,\bar{\ell }_i^{(k)})})\right) \nonumber \\&\quad +{M}\left( \sum _{i=1}^p\sum _{\ell =0}^{{L_i^{(k)}-1}}\Vert x_i^{(k,\ell +1)}-x_i^{(k,\ell )}\Vert +\Vert x_i^{(k,\bar{\ell }_i^{(k)})}-\hat{u}_i^{(k,\bar{\ell }_i^{(k)})}\Vert \right) ^2\nonumber \\&\quad +\frac{\mu }{2\alpha _{\min }}\sum _{i=1}^{p}\left( \sum _{\ell =0}^{\bar{\ell }_i^{(k)}-1}\Vert x_i^{(k,\ell +1)}-x_i^{{(k,\ell )}}\Vert \right) ^2\nonumber \\&\quad +\frac{{M}}{2}\sum _{i=1}^{p}\left( \sum _{\ell =0}^{\bar{\ell }_i^{(k)}-1}\Vert x_i^{(\ell +1)}-x_i^{(k,\ell )}\Vert +\Vert x_i^{(k,\bar{\ell }_i^{(k)})}-\hat{u}_i^{(k,\bar{\ell }_i^{(k)})}\Vert \right) ^2. \end{aligned}$$

(B14)

By combining Lemma 5 with Lemma 7, we obtain the following limits as $k\rightarrow \infty$

$$\begin{aligned} \lim _{k\rightarrow \infty }\sum _{i=1}^{p}\left( -h_i^{(k,\bar{\ell }_i^{(k)})}(u_i^{(k,\bar{\ell }_i^{(k)})})\right)&=0\\ \lim _{k\rightarrow \infty }\sum _{\ell =0}^{\bar{\ell }_i^{(k)}-1}\Vert x_i^{(k,\ell +1)}-x_i^{(k,\ell )}\Vert +\Vert x_i^{(k,\bar{\ell }_i^{(k)})}-\hat{u}_i^{(k,\bar{\ell }_i^{(k)})}\Vert&=0\\ \lim _{k\rightarrow \infty }\sum _{\ell =0}^{L_i^{(k)}-1}\Vert x_i^{(k,\ell +1)}-x_i^{(k,\ell )}\Vert +\Vert x_i^{(k,\bar{\ell }_i^{(k)})}-\hat{u}_i^{(k,\bar{\ell }_i^{(k)})}\Vert&=0. \end{aligned}$$

Hence inequality (B14) yields the thesis. $\square$