**Q4.**
Yes, we can extract an $O(1/k)$ convergence rate from our proven inequalities.
We state and prove a more generalized statement here.
The assumption is the same as Lemma G.1 (which covers the assumption of Lemma 4.1), and we restate it here for convenience.
**Theorem.**
Assume $f\colon \mathbf{R}^m \to \mathbf{R} \cup \\{ \infty \\}$ is a strictly convex function
and the dynamic interconnect is admissible.
Let a discrete-time optimization algorithm generate a sequence $\\{(v^k, i^k, x^k, y^k)\\}^{\infty}\_{k=0}$.
Suppose there exists
$\eta > 0$ such that for all $k=0, 1, \ldots$ the energy descent
$$
\mathcal{E}\_{k+1} + \eta \langle x^k - x^\star, y^k - y^\star \rangle - \mathcal{E}\_k \leq 0
$$
holds.
Then, for the Lagrangian defined as $L(x, z, y) = f(x) - y^T(x - E^\intercal z)$, we have
$$
\min\_{k \in \\{0, 1, \dots, K\\}} \left( L(x^k, z^\star, y^\star) - f(x^\star) \right)
\leq \frac{1}{(K+1)\eta} \mathcal{E}\_0
= O\left(\frac{1}{K}\right)
$$
for $K=0,1,\dots$.
*Proof outline.*
Since we are considering the same assumption as in Lemma G.1, all arguments used in its proof are applicable.
From line 1223 we have
$$
0 \leq \sum\_{k=0}^K \eta \langle x^k - x^\star, y^k - y^\star \rangle \leq \mathcal{E}\_0,
$$
and from line 1229 we have
$$
\langle x^k-x^\star, y^k-y^\star\rangle \geq L(x^k, z^\star, y^\star) - f(x^\star) \geq 0.
$$
Combining two inequalities and dividing both sides by $K+1$, we get the desired conclusion. $\blacksquare$
All algorithms obtained by our framework, including DADMM+C, achieve this convergence rate. As mentioned in an earlier part of this response, doing a more refined analysis of convergence rates (establishing, say, linear rates of convergence in an automated fashion) is a follow-up direction of work that we are pursuing.
**Q5.**
Explicit knowledge of $v^\star$ and $i^\star$ is not needed.
We can find the numerical values of $(\alpha,\beta,h)$ without explicit knowledge of the optimal values.
For example, consider finding dissipative discretization for gradient descent.
Let $f$ be $L$-smooth.
The iterates are given by $x^{k+1} = x^k- h\nabla f(x^k)$.
The energy is $\mathcal{E}\_k = \frac{1}{2}\\|x^k-x^\star\\|^2\_2$, where $x^\star$ is such that
$y^\star = \nabla f(x^\star) = 0$.
Then we want to find step size $h>0$ and $\eta>0$ such that
$$\mathcal{E}\_{k+1}+\eta\langle x^k - x^\star, y^k - y^\star\rangle-\mathcal{E}\_k \leq 0.$$
Using the inequality arises from $L$-smoothness of $f$,
$$
\langle x^k - x^\star, y^k - y^\star\rangle
\geq \frac{1}{L} \\|y^k \\|^2\_2,
$$
we can check that the discretization is dissipative for all
$\eta < \frac{1}{2L}$
and $h \in [\frac{1}{L} \pm \frac{1}{L}\sqrt{1-2\eta L}]$.
Thus, we can find proper $\eta$ and $h$ without explicitly specifying $x^\star$, by just representing it as a point satisfying the optimality conditions. Further details on this line of reasoning are available in the PEP papers [1, 2].
[1] Y. Drori and M. Teboulle. Performance of first-order methods for smooth convex minimization: A novel approach. *Mathematical Programming*, 145(1):451–482, 2014.
[2] A. B. Taylor, J. M. Hendrickx, and F. Glineur. Smooth strongly convex interpolation and exact worst-case performance of first-order methods. *Mathematical Programming*, 161(1):307–345, 2017.
**Q6.**
One approach is to let the solver find the RLC values along with the discretization. (We recently implemented this functionality in ciropt.)
Another approach is to try a few variations for the values for resistance, inductance and
capacitance, and see for which of
them the solver finds a discretization.
### **Questions: Minor issues**
**Q1.**
We clarify that this is not a mistake. For the circuit in Figure 2, for example, there are $\tau=8$ nodes, but only $m=5$ of the nodes (node $1, 2, \dots 5$) are connected to the terminals. Likewise, there are nodes that are connected to neither the terminal nor the ground. The potential of those nodes is denoted by $e$.
**Q2.**
Thank you for pointing out to us this typo; we will correct it.
We appreciate the reviewer’s thoughtful questions. We hope we have adequately addressed the reviewer's concern regarding the extra computation related to the performance estimation problem and the possibility of obtaining convergence rates from our approach. If so, we kindly ask the reviewer to consider raising the score.