Iteration Complexity Analysis of Multi-block ADMM for a Family of Convex Minimization Without Strong Convexity

Abstract

The alternating direction method of multipliers (ADMM) is widely used in solving structured convex optimization problems due to its superior practical performance. On the theoretical side however, a counterexample was shown in Chen et al. (Math Program 155(1):57–79, 2016.) indicating that the multi-block ADMM for minimizing the sum of N $(N\ge 3)$ convex functions with N block variables linked by linear constraints may diverge. It is therefore of great interest to investigate further sufficient conditions on the input side which can guarantee convergence for the multi-block ADMM. The existing results typically require the strong convexity on parts of the objective. In this paper, we provide two different ways related to multi-block ADMM that can find an $\epsilon$-optimal solution and do not require strong convexity of the objective function. Specifically, we prove the following two results: (1) the multi-block ADMM returns an $\epsilon$-optimal solution within $O(1/\epsilon ^2)$ iterations by solving an associated perturbation to the original problem; this case can be seen as using multi-block ADMM to solve a modified problem; (2) the multi-block ADMM returns an $\epsilon$-optimal solution within $O(1/\epsilon )$ iterations when it is applied to solve a certain sharing problem, under the condition that the augmented Lagrangian function satisfies the Kurdyka–Łojasiewicz property, which essentially covers most convex optimization models except for some pathological cases; this case can be seen as applying multi-block ADMM to solving a special class of problems.

Appendix: Proof of Theorem 4.3

We first prove the following lemma.

Lemma 4.8

The following results hold under the conditions in Scenario 2.

• 1. The iterative gap of dual variable can be bounded by that of primal variable, i.e.,

  $$\begin{aligned} \Vert \lambda ^{k+1} - \lambda ^{k} \Vert ^2 \le L^2 \Vert x_N^{k+1} - x_N^k \Vert , \end{aligned}$$

  (4.17)

  where L is the Lipschitz constant for $\nabla f_N$.

• 2. The augmented Lagrangian $L_\gamma$ has a sufficient decrease in each iteration, i.e.,

  $$\begin{aligned}&{\mathcal {L}}_\gamma \left( x_1^{k},\ldots , x_{N+1}^{k};\lambda ^k\right) - {\mathcal {L}}_\gamma \left( x_1^{k+1},\ldots , x_{N+1}^{k+1};\lambda ^{k+1}\right) \nonumber \\&\quad \ge \frac{\gamma ^2-2L^2}{2\gamma (1+L^2)}\left( \sum \limits _{i=1}^{N-1} \left\| A_i x_i^k - A_i x_i^{k+1} \right\| ^2 + \left\| x_N^k - x_N^{k+1} \right\| ^2 + \left\| \lambda ^k - \lambda ^{k+1} \right\| ^2\right) .\nonumber \\ \end{aligned}$$

  (4.18)

• 3. The augmented Lagrangian ${\mathcal {L}}_\gamma (w^k)$ is uniformly lower bounded, and it holds true that

  $$\begin{aligned}&\sum \limits _{k=0}^\infty \left( \sum \limits _{i=1}^{N-1} \left\| A_i x_i^{k+1} - A_i x_i^k \right\| ^2 + \left\| x_N^{k+1} - x_N^k \right\| ^2 + \left\| \lambda ^{k+1}-\lambda ^k \right\| ^2\right) \nonumber \\&\quad \le \frac{2\gamma (1+L^2)}{\gamma ^2-2L^2}\left( {\mathcal {L}}_\gamma (w^0) - L^*\right) \end{aligned}$$

  (4.19)

  where $L^*$ is the uniform lower bound of ${\mathcal {L}}_\gamma (w^k)$, and hence

  $$\begin{aligned} \lim \limits _{k\rightarrow \infty } \left( \sum _{i=1}^{N-1} \left\| A_i x_i^k - A_i x_i^{k+1} \right\| ^2 + \left\| x_N^k - x_N^{k+1} \right\| ^2 + \left\| \lambda ^k - \lambda ^{k+1} \right\| ^2 \right) = 0. \end{aligned}$$

  (4.20)

  Moreover, $\left\{ \left( x_1^k, x_2^k, \ldots ,x_N^k, \lambda ^k\right) : k=0,1,\ldots \right\}$ is a bounded sequence.

• 4. There exists an upper bound for a subgradient of augmented Lagrangian ${\mathcal {L}}_\gamma$ in each iteration. In fact, we define

  $$\begin{aligned} R_i^{k+1}:= & {} \gamma A_i^\top \left( \sum \limits _{i=1}^{N-1} A_i x_i^{k+1} + x_N^{k+1} - b\right) \\&- \gamma A_{i}^{\top }\left( \sum \limits _{j=i+1}^{N-1} A_{j}\left( x_{j}^{k}-x_{j}^{k+1}\right) +\left( x_{N}^{k}-x_{N}^{k+1}\right) \right) \end{aligned}$$

  and

  $$\begin{aligned} R_N^{k+1} := \gamma \left( \sum \limits _{i=1}^{N-1} A_i x_i^{k+1} + x_N^{k+1} - b\right) , \quad R_{\lambda }^{k+1} := b - \sum \limits _{i=1}^{N-1} A_i x_i^{k+1} - x_N^{k+1} \end{aligned}$$

  for each positive integer k, and $i = 1,2,\ldots , N$. Then $\left( R_1^{k+1}, \ldots , R_N^{k+1}, R_\lambda ^{k+1}\right) \in \partial {\mathcal {L}}_\gamma (w^{k+1})$. Moreover, it holds that

  $$\begin{aligned}&\left\| \left( R_1^{k+1}, \ldots , R_N^{k+1}, R_\lambda ^{k+1}\right) \right\| \le \sum \limits _{i=1}^N \left\| R_i^{k+1} \right\| + \left\| R_\lambda ^{k+1} \right\| \nonumber \\&\quad \le M\left( \sum \limits _{i=1}^{N-1} \left\| A_i x_i^k - A_i x_i^{k+1}\right\| + \left\| x_i^k - x_i^{k+1} \right\| + \left\| \lambda ^k - \lambda ^{k+1} \right\| \right) , \quad \forall k\ge 0,\nonumber \\ \end{aligned}$$

  (4.21)

  where M is a constant defined in (4.8).

Proof of Lemma 4.8

• 1. (4.17) follows directly from (4.5).

• 2. From (4.1), by invoking the convexity of $f_i$, we have for $i=1,\ldots ,N-1$:

  $$\begin{aligned} 0&= \left( x_i^k - x_i^{k+1}\right) ^\top \left[ g_i\left( x_{i}^{k+1}\right) -A_{i}^{\top }\lambda ^{k}+\gamma A_{i}^{\top }\left( \sum _{j=1}^iA_{j}x_{j}^{k+1}+\sum _{j=i+1}^{N-1} A_{j}x_{j}^{k} + x_{N}^k -b\right) \right] \nonumber \\&\le f_i\left( x_i^k\right) - f_i\left( x_i^{k+1}\right) - \left( A_i x_i^k - A_i x_i^{k+1}\right) ^\top \lambda ^k \nonumber \\&\qquad +\, \gamma \left( A_i x_i^k - A_i x_i^{k+1}\right) ^\top \left( \sum _{j=1}^iA_{j}x_{j}^{k+1}+\sum _{j=i+1}^{N-1} A_{j}x_{j}^{k} + x_{N}^k -b \right) \nonumber \\&= {\mathcal {L}}_\gamma \left( x_1^{k+1},\ldots , x_{i-1}^k,x_i^k,\ldots ,\lambda ^k\right) - {\mathcal {L}}_\gamma \left( x_1^{k+1},\ldots ,x_i^{k+1},x_{i+1}^k,\ldots ,\lambda ^k\right) \nonumber \\&\qquad -\, \frac{\gamma }{2}\left\| A_i x_i^k - A_i x_i^{k+1}\right\| ^2. \end{aligned}$$

  (4.22)

  Similarly, from (4.2) we can prove that

  $$\begin{aligned} {\mathcal {L}}_\gamma \left( x_1^{k+1},\ldots , x_{N-1}^{k+1},x_N^k,\ldots ;\lambda ^k\right) - {\mathcal {L}}_\gamma \left( x_1^{k+1},\ldots ,x_N^{k+1};\lambda ^k\right) \ge \frac{\gamma }{2}\left\| x_N^k - x_N^{k+1}\right\| ^2.\nonumber \\ \end{aligned}$$

  (4.23)

  Summing (4.22) over $i=1,\ldots ,N-1$ and (4.23), we have

  $$\begin{aligned}&{\mathcal {L}}_\gamma \left( x_1^k,\ldots ,x_N^k,\lambda ^k\right) - {\mathcal {L}}_\gamma \left( x_1^{k+1},\ldots , x_N^{k+1},\lambda ^k\right) \nonumber \\&\quad \ge \frac{\gamma }{2}\sum \limits _{i=1}^{N-1}\left\| A_i x_i^k - A_i x_i^{k+1} \right\| ^2 + \frac{\gamma }{2}\left\| x_N^k - x_N^{k+1} \right\| ^2. \end{aligned}$$

  (4.24)

  On the other hand, it follows from (4.2) and (4.17) that

  $$\begin{aligned}&{\mathcal {L}}_\gamma \left( x_1^{k+1},\ldots , x_N^{k+1},\lambda ^k\right) - {\mathcal {L}}_\gamma \left( x_1^{k+1},\ldots , x_N^{k+1},\lambda ^{k+1}\right) \nonumber \\&\quad = \frac{1}{\gamma }\left\| \lambda ^k - \lambda ^{k+1} \right\| ^2 \ge - \frac{L^2}{\gamma }\left\| x_N^k - x_N^{k+1} \right\| ^2. \end{aligned}$$

  (4.25)

  Combining (4.24) and (4.25) yields

  $$\begin{aligned}&{\mathcal {L}}_\gamma \left( x_1^k,\ldots , x_N^k,\lambda ^k\right) - {\mathcal {L}}_\gamma \left( x_1^{k+1},\ldots , x_N^{k+1},\lambda ^{k+1}\right) \nonumber \\&\quad \ge \frac{\gamma }{2}\sum \limits _{i=1}^{N-1}\left\| A_i x_i^k - A_i x_i^{k+1} \right\| ^2 + \frac{\gamma ^2 - 2L^2}{2\gamma } \left\| x_N^k - x_N^{k+1} \right\| ^2 \nonumber \\&\quad \ge \frac{\gamma }{2}\sum \limits _{i=1}^{N-1}\left\| A_i x_i^k - A_i x_i^{k+1} \right\| ^2 + \frac{\gamma ^2 - 2L^2}{2\gamma (1+L^2)}\left( \left\| x_N^k - x_N^{k+1} \right\| ^2 + \left\| \lambda ^k - \lambda ^{k+1} \right\| ^2\right) \nonumber \\&\quad \ge \frac{\gamma ^2 - 2L^2}{2\gamma (1+L^2)}\left( \sum \limits _{i=1}^{N-1}\left\| A_i x_i^k - A_i x_i^{k+1} \right\| ^2 + \left\| x_N^k - x_N^{k+1} \right\| ^2 + \left\| \lambda ^k - \lambda ^{k+1} \right\| ^2\right) .\nonumber \\ \end{aligned}$$

  (4.26)

• 3. It follows from (4.2) and the fact that $\nabla f_N$ is Lipschitz continuous with constant L that,

  $$\begin{aligned}&f_N\left( b - \sum \limits _{i=1}^{N-1} A_i x_i^{k+1}\right) \\&\quad \le f_N\left( x_N^{k+1}\right) + \left\langle \nabla f_N\left( x_N^{k+1}\right) , \left( b - \sum \limits _{i=1}^{N-1} A_i x_i^{k+1} - x_N^{k+1}\right) \right\rangle \\&\qquad +\, \frac{L}{2}\left\| b - \sum \limits _{i=1}^{N-1} A_i x_i^{k+1} - x_N^{k+1} \right\| ^2 \\&\quad = f_N\left( x_N^{k+1}\right) - \left\langle \lambda ^{k+1}, \sum \limits _{i=1}^{N-1} A_i x_i^{k+1} + x_N^{k+1} - b\right\rangle + \frac{L}{2} \left\| \sum \limits _{i=1}^{N-1} A_i x_i^{k+1} + x_N^{k+1} - b\right\| ^2. \end{aligned}$$

  This implies that there exists $L^*>-\infty$, such that

  $$\begin{aligned}&{\mathcal {L}}_\gamma \left( x_1^{k+1},\ldots , x_N^{k+1},\lambda ^{k+1}\right) \nonumber \\&\quad \ge \sum _{i=1}^{N-1} f_i(x_i^{k+1}) + f_N\left( b - \sum \limits _{i=1}^{N-1} A_i x_i^{k+1}\right) + \frac{\gamma -L}{2} \left\| \sum _{i=1}^{N-1} A_i x_i^{k+1} + x_N^{k+1} -b\right\| ^2 \nonumber \\&\quad > L^*, \end{aligned}$$

  (4.27)

  where the last inequality holds since $\gamma >L$ and $\inf _{{\mathcal {X}}_i}f_i>f_i^*$ for $i=1,2,\ldots ,N$.

  Therefore, it directly follows from (4.18) and $\gamma >\sqrt{2}L$ that,

  $$\begin{aligned}&\frac{\gamma ^2-2L^2}{2\gamma (1+L^2)}\sum \limits _{k=0}^K \left( \sum \limits _{i=1}^{N-1} \Vert A_i x_i^{k+1} - A_i x_i^k \Vert ^2 + \Vert x_N^{k+1} - x_N^k \Vert ^2 + \Vert \lambda ^{k+1}-\lambda ^k\Vert ^2\right) \\&\quad \le {\mathcal {L}}_\gamma (w^0) - L^*. \end{aligned}$$

  Letting $K\rightarrow \infty$ gives (4.19) and (4.20).

  It also follows from (4.27), (4.18) and $\gamma >\sqrt{2}L$ that ${\mathcal {L}}_\gamma (w^0) - f_N^* \ge \sum _{i=1}^{N-1} f_i(x_i^{k+1})$. This implies that $\left\{ \left( x_1^k, x_2^k, \ldots ,x_{N-1}^k\right) : k=0,1,\ldots \right\}$ is a bounded sequence by using the coerciveness of $f_i+\mathbf 1 _{{\mathcal {X}}_i}, i=1,2,\ldots ,N-1$. The boundedness of $\left( x_N^k, \lambda ^k\right)$ can be obtained by using (4.3), (4.17) and (4.20).

• 4. From the definition of ${\mathcal {L}}_\gamma$, it is clear that for $i=1,\ldots ,N-1$,

  $$\begin{aligned} g_i\left( x_i^{k+1}\right) - A_i^\top \lambda ^{k+1} + \gamma A_i^\top \left( \sum \limits _{i=1}^{N-1} A_i x_i^{k+1} + x_N^{k+1} - b\right) \in \partial _{x_i} {\mathcal {L}}_\gamma (w^{k+1}), \end{aligned}$$

  and

  $$\begin{aligned} \nabla f\left( x_N^{k+1}\right) - \lambda ^{k+1} + \gamma \left( \sum \limits _{i=1}^{N-1} A_i x_i^{k+1} + x_N^{k+1} - b\right) = \nabla _{x_N} {\mathcal {L}}_\gamma (w^{k+1}), \end{aligned}$$

  and

  $$\begin{aligned} b - \sum \limits _{i=1}^{N-1} A_i x_i^{k+1} - x_N^{k+1} = \nabla _{\lambda } {\mathcal {L}}_\gamma (w^{k+1}), \end{aligned}$$

  where $g_i\in \partial \left( f_i + \mathbf {1}_{{\mathcal {X}}_i}\right)$ for $i=1,2,\ldots ,N-1$.

  Combining these relations with (4.4) and (4.5) yields that

  $$\begin{aligned}&R_i^{k+1} := \gamma A_i^\top \left( \sum \limits _{i=1}^{N-1} A_i x_i^{k+1} + x_N^{k+1} - b\right) \\&\quad - \gamma A_{i}^{\top }\left( \sum \limits _{j=i+1}^{N-1} A_{j}(x_{j}^{k}-x_{j}^{k+1})+(x_{N}^{k}-x_{N}^{k+1})\right) \in \partial _{x_i} {\mathcal {L}}_\gamma (w^{k+1}), \\&R_N^{k+1} := \gamma \left( \sum \limits _{i=1}^{N-1} A_i x_i^{k+1} + x_N^{k+1} - b\right) = \nabla _{x_N} {\mathcal {L}}_\gamma (w^{k+1}), \\&R_{\lambda }^{k+1} := b - \sum \limits _{i=1}^{N-1} A_i x_i^{k+1} - x_N^{k+1} = \nabla _{\lambda } {\mathcal {L}}_\gamma (w^{k+1}), \end{aligned}$$

  for $i=1,2,\ldots ,N-1$. Therefore, $\left( R_1^{k+1}, \ldots , R_N^{k+1}, R_\lambda ^{k+1}\right) \in \partial {\mathcal {L}}_\gamma (w^{k+1})$. We now need to bound the norms of $R_i^{k+1}$, $i=1,\ldots ,N-1$, $R_N^k$ and $R_\lambda ^k$. It holds that

  $$\begin{aligned} \left\| R_i^{k+1} \right\|\le & {} \gamma \left\| A_i^\top \right\| \left( \sum \limits _{j=i+1}^{N-1} \left\| A_j x_j^{k} - A_j x_j^{k+1}\right\| + \left\| x_N^{k} - x_N^{k+1}\right\| \right) \\&+\, \gamma \left\| A_i^\top \right\| \left\| \sum \limits _{i=1}^{N-1} A_i x_i^{k+1} + x_N^{k+1} - b\right\| \\\le & {} \gamma \left\| A_i^\top \right\| \left( \sum \limits _{j=1}^{N-1} \left\| A_j x_j^{k} - A_j x_j^{k+1}\right\| + \left\| x_N^{k} - x_N^{k+1}\right\| \right) + \left\| A_i^\top \right\| \left\| \lambda ^k - \lambda ^{k+1}\right\| \end{aligned}$$

  and

  $$\begin{aligned} \left\| R_N^{k+1} \right\| = \gamma \left\| \sum \limits _{i=1}^{N-1} A_i x_i^{k+1} + x_N^{k+1} - b\right\| = \left\| \lambda ^k - \lambda ^{k+1}\right\| , \quad \Vert R_\lambda ^{k+1} \Vert = \frac{1}{\gamma } \left\| \lambda ^k - \lambda ^{k+1} \right\| . \end{aligned}$$

  These relations immediately imply (4.21). $\Box$

Proof of Theorem 4.3

• 1. It has been proven in Lemma 4.8 that $\left\{ \left( x_1^k, x_2^k, \ldots ,x_N^k, \lambda ^k\right) :\right. \left. k=0,1,\ldots \right\}$ is a bounded sequence. Therefore, we conclude that $\Omega (w^0)$ is non-empty by the Bolzano-Weierstrass Theorem. Let $w^* = \left( x_1^*,\ldots , x_N^*,\lambda ^*\right) \in \Omega (w^0)$ be a limit point of $\{w^k = \left( x_1^k,\ldots , x_N^k,\lambda ^k\right) :k=0,1,\ldots \}$. Then there exists a subsequence $\left\{ w^{k_q} = \left( x_1^{k_q},\ldots , x_N^{k_q},\lambda ^{k_q}\right) :q=0,1,\ldots \right\}$ such that $w^{k_q}\rightarrow w^*$ as $q\rightarrow \infty$. Since $f_i, i=1,\ldots ,N-1$, are lower semi-continuous, we obtain that

  $$\begin{aligned} \liminf \limits _{q\rightarrow \infty } f_i(x_i^{k_q}) \ge f_i(x_i^*), \quad i=1,2,\ldots , N. \end{aligned}$$

  (4.28)

  From (1.2), we have for any integer k and any $i=1,\ldots ,N-1$,

  $$\begin{aligned} x_i^{k+1} := \mathop {\mathrm{argmin}}\limits _{x_i\in {\mathcal {X}}_i} \ {\mathcal {L}}_\gamma \left( x_1^{k+1},\ldots , x_{i-1}^{k+1}, x_i, x_{i+1}^k,\ldots , x_N^k;\lambda ^k\right) . \end{aligned}$$

  Letting $x_i = x_i^*$ in the above, we get

  $$\begin{aligned} {\mathcal {L}}_\gamma \left( x_1^{k+1},\ldots , x_i^{k+1}, x_{i+1}^k,\ldots , x_N^k;\lambda ^k\right) \le {\mathcal {L}}_\gamma \left( x_1^{k+1},\ldots , x_{i-1}^{k+1}, x_i^{*}, x_{i+1}^k,\ldots , x_N^k;\lambda ^k\right) , \end{aligned}$$

  i.e.,

  $$\begin{aligned}&f_i\left( x_i^{k+1}\right) - \left\langle \lambda ^k, A_i x_i^{k+1}\right\rangle + \frac{\gamma }{2}\left\| \sum \limits _{j=1}^i A_j x_j^{k+1} + \sum \limits _{j=i+1}^{N-1} A_j x_j^{k} + x_N^k - b \right\| ^2 \\&\quad \le f_i\left( x_i^*\right) - \left\langle \lambda ^k, A_i x_i^*\right\rangle + \frac{\gamma }{2}\left\| \sum \limits _{j=1}^{i-1} A_j x_j^{k+1} + A_i x_i^* + \sum \limits _{j=i+1}^{N-1} A_j x_j^{k} + x_N^k - b \right\| ^2. \end{aligned}$$

  Choosing $k=k_q-1$ in the above inequality and letting q go to $+\infty$, we obtain

  $$\begin{aligned} \limsup \limits _{q\rightarrow +\infty }f_i\left( x_i^{k_q}\right) \le \limsup \limits _{q\rightarrow +\infty } \left( \frac{\gamma }{2}\left\| A_i x_i^{k_q} - A_i x_i^* \right\| ^2 + \left\langle \lambda ^k, A_i x_i^{k_q}- A_i x_i^*\right\rangle \right) + f_i\left( x_i^*\right) ,\nonumber \\ \end{aligned}$$

  (4.29)

  for $i=1,2,\ldots ,N-1$. Here we have used the facts that the sequence $\{w^k:k=0,1,\ldots \}$ is bounded, and $\gamma$ is finite, and that the distance between two successive iterates tends to zero (4.20), and the fact that

  $$\begin{aligned}&\sum \limits _{j=1}^i A_j x_j^{k+1} + \sum \limits _{j=i+1}^{N-1} A_j x_j^k + x_N^k - b = \sum \limits _{j=i+1}^{N-1} \left( A_j x_j^k - A_j x_j^{k+1}\right) + \left( x_N^k - x_N^{k+1}\right) \\&\quad +\,\frac{1}{\gamma } \left( \lambda ^k - \lambda ^{k+1}\right) . \end{aligned}$$

  From (4.20) we also have $x_i^{k_q-1}\rightarrow x_i^*$ as $q\rightarrow \infty$, hence (4.29) reduces to $\limsup \limits _{q\rightarrow \infty }f_i(x_i^{k_q})\le f_i(x_i^*)$. Therefore, combining with (4.28), $f_i(x_i^{k_q})$ tends to $f_i(x_i^*)$ as $q\rightarrow \infty$. Hence, we can conclude that

  $$\begin{aligned} \lim \limits _{q\rightarrow \infty }{\mathcal {L}}_\gamma (w^{k_q})= & {} \lim \limits _{q\rightarrow \infty }\left( \sum \limits _{i=1}^N f_i\left( x_i^{k_q}\right) - \left\langle \lambda ^{k_q}, \sum \limits _{i=1}^{N-1} A_i x_i^{k_q} + x_N^{k_q} -b\right\rangle \right. \\&\left. +\, \frac{\gamma }{2}\left\| \sum \limits _{i=1}^{N-1} A_i x_i^{k_q} + x_N^{k_q} -b \right\| ^2\right) \\= & {} \sum \limits _{i=1}^N f_i\left( x_i^{*}\right) - \left\langle \lambda ^{*}, \sum \limits _{i=1}^{N-1} A_i x_i^{*}+x_N^{*}-b\right\rangle + \frac{\gamma }{2}\left\| \sum \limits _{i=1}^{N-1} A_i x_i^{*}+x_N^{*}-b \right\| ^2 \\= & {} {\mathcal {L}}_\gamma (w^{*}). \end{aligned}$$

  On the other hand, it follows from (4.20) and (4.21) that

  $$\begin{aligned} \left( R_1^{k+1}, \ldots , R_N^{k+1}, R_\lambda ^{k+1}\right)\in & {} \partial {\mathcal {L}}_\gamma (w^{k+1}) \end{aligned}$$

  (4.30)
  $$\begin{aligned} \left( R_1^{k+1}, \ldots , R_N^{k+1}, R_\lambda ^{k+1}\right)\rightarrow & {} (0,\ldots ,0), \quad k\rightarrow \infty . \end{aligned}$$

  (4.31)

  It implies that $(0,\ldots ,0)\in \partial {\mathcal {L}}_\gamma (x_1^*,\ldots ,x_N^*,\lambda ^*)$ due to the closeness of $\partial {\mathcal {L}}_\gamma$. Therefore, $w^* = \left( x_1^*,\ldots ,x_N^*,\lambda ^*\right)$ is a critical point of ${\mathcal {L}}_\gamma (x_1,\ldots ,x_N,\lambda )$.

• 2. The proof for this assertion directly follows from Lemma 5 and Remark 5 of [4]. We omit the proof here for succinctness.

• 3. We define that $\tilde{L}$ is the finite limit of ${\mathcal {L}}_\gamma (x_1^k,\ldots ,x_N^k,\lambda ^k)$ as k goes to infinity, i.e.,

  $$\begin{aligned} \tilde{L} = \lim \limits _{k\rightarrow \infty } {\mathcal {L}}_\gamma \left( x_1^k,\ldots ,x_N^k,\lambda ^k\right) . \end{aligned}$$

  Take $w^*\in \Omega (w^0)$. There exists a subsequence $w^{k_q}$ converging to $w^*$ as q goes to infinity. Since we have proven that

  $$\begin{aligned} \lim \limits _{q\rightarrow \infty }{\mathcal {L}}_\gamma (w^{k_q}) = {\mathcal {L}}_\gamma (w^{*}), \end{aligned}$$

  and ${\mathcal {L}}_\gamma (w^{k})$ is a non-increasing sequence, we conclude that ${\mathcal {L}}_\gamma (w^{*}) = \tilde{L}$, hence the restriction of ${\mathcal {L}}_\gamma (x_1,\ldots ,x_N,\lambda )$ to $\Omega (w^0)$ equals $\tilde{L}$.

$\square$