---
abstract: 'This paper investigates the role of feasible initial guesses and large consensus-violation penalization in distributed optimization for Optimal Power Flow ([<span style="font-variant:small-caps;">opf</span>]{}) problems.  Specifically, we discuss the behavior of the Alternating Direction of Multipliers Method ([<span style="font-variant:small-caps;">admm</span>]{}).  We show that in case of large consensus-violation penalization [<span style="font-variant:small-caps;">admm</span>]{}might exhibit slow progress.  We support this observation by an analysis of the algorithmic properties of [<span style="font-variant:small-caps;">admm</span>]{}.  Furthermore, we illustrate our findings considering the [<span style="font-variant:small-caps;">ieee</span>]{}57 bus system and we draw upon a comparison of [<span style="font-variant:small-caps;">admm</span>]{}and the Augmented Lagrangian Alternating Direction Inexact Newton ([<span style="font-variant:small-caps;">aladin</span>]{}) method.' 
author:
- 'Alexander Engelmann$^\star$'
- 'Timm Faulwasser$^\star$'
bibliography:
- 'paper.bib'
title: |
    Feasibility vs. Optimality in Distributed AC OPF—\
    A Case Study Considering ADMM and ALADIN [^1] 
---

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Introduction
============

Distributed [optimization algorithms]{} for [<span style="font-variant:small-caps;">ac</span>]{}Optimal Power Flow ([<span style="font-variant:small-caps;">opf</span>]{}) recently gained significant interest as these problems are inherently non-convex and often large-scale; i.e. comprising up to several thousand buses [@Guo2017].  Distributed optimization is considered to be helpful as it allows splitting large [<span style="font-variant:small-caps;">opf</span>]{}problems into several smaller subproblems; thus reducing complexity and avoiding the exchange of full grid models between subsystems.  We refer to [@Molzahn2017] for a recent overview of distributed optimization and control approaches in power systems. 

One frequently discussed *convex* distributed optimization method is the Alternating Direction of Multipliers Method ([<span style="font-variant:small-caps;">admm</span>]{}) [@Boyd2011], which is also applied in context of [<span style="font-variant:small-caps;">ac</span>]{}[<span style="font-variant:small-caps;">opf</span>]{}[@Guo2017; @Erseghe2014a; @Kim2000]. [ <span style="font-variant:small-caps;">admm</span>]{}often yields promising results even for large-scale power systems [@Erseghe2014a].  However, [<span style="font-variant:small-caps;">admm</span>]{}sometimes requires a specific partitioning technique and/or a feasible initialization in combination with high consensus-violation penalization parameters to converge [@Guo2017].  The requirement of feasible initialization seems quite limiting as it requires solving a centralized inequality-constrained power flow problem requiring full topology and load information leading to approximately the same complexity as full [<span style="font-variant:small-caps;">opf</span>]{}. 

In previous works [@Engelmann2019; @Engelmann18a; @Murray2018] we suggested applying the Augmented Lagrangian Alternating Direction Inexact Newton ([<span style="font-variant:small-caps;">aladin</span>]{}) method to stochastic and deterministic [<span style="font-variant:small-caps;">opf</span>]{}problems ranging from 5 to 300 buses.  In case a certain line-search is applied [@Houska2016], [<span style="font-variant:small-caps;">aladin</span>]{}provides global convergence guarantees to local minimizers for non-convex problems without the need of feasible initialization.  The results in [@Engelmann2019] underpin that [<span style="font-variant:small-caps;">aladin</span>]{}is able to outperform [<span style="font-variant:small-caps;">admm</span>]{}in many cases.  This comes at cost of a higher per-step information exchange compared with [<span style="font-variant:small-caps;">admm</span>]{}and a more complicated coordination step, cf. [ @Engelmann2019]. 

In this paper we investigate the interplay of feasible initialization with high penalization for consensus violation in [<span style="font-variant:small-caps;">admm</span>]{}for distributed [<span style="font-variant:small-caps;">ac</span>]{}[<span style="font-variant:small-caps;">opf</span>]{}.  We illustrate our findings on the [<span style="font-variant:small-caps;">ieee</span>]{}57-bus system.  Furthermore, we compare the convergence behavior of [<span style="font-variant:small-caps;">admm</span>]{}to [<span style="font-variant:small-caps;">aladin</span>]{}not suffering from the practical need for feasible initialization [@Houska2016].  Finally, we provide theoretical results supporting our numerical observations for the convergence behavior of [<span style="font-variant:small-caps;">admm</span>]{}. 

The paper is organized as follows: In Section \[sec:ALADINandADMM\] we briefly recap [<span style="font-variant:small-caps;">admm</span>]{}and [<span style="font-variant:small-caps;">aladin</span>]{}including their convergence properties.  Section \[sec:NumRes\] shows numerical results for the [<span style="font-variant:small-caps;">ieee</span>]{}57-bus system focusing on the influence of the penalization parameter $\rho$ on the convergence behavior of [<span style="font-variant:small-caps;">admm</span>]{}.  Section \[sec:Theory\] presents an analysis of the interplay between high penalization and a feasible initialization. 

ALADIN and ADMM {#sec:ALADINandADMM}
===============

[For distributed optimization, [<span style="font-variant:small-caps;">opf</span>]{}problems are often formulated in *affinely coupled separable form* $$\label{eq:OPF}
        \min_{x\in\mathbb{R}^{n_x}} \, \sum_{i\in \mathcal{R}} f_i(x_i) \quad\text{subject to}\quad x_i \in \mathcal{X}_i,\; \forall\, i \in \mathcal{R} \quad\text{and}\quad
    \sum_{i\in \mathcal{R}}A_i x_i=0,$$ where the decision vector is divided into sub-vectors $x^\top=[x_1^\top,\dots,x_{|\mathcal{R}|}^\top] \in \mathbb{R}^{n_x}$, $\mathcal{R}$ is the index set of subsystems usually representing geographical areas of a power system and local nonlinear constraint sets $\mathcal{X}_i:=\{x_i \in \mathbb{R}^{n_{xi}} \; | \; h_{i}(x_i) \leq 0\}$. ]{}Througout this work we assume that $f_i$ and $h_i$ are twice continuously differentialble and that all $\mathcal{X}_i$ are compact.  Note that the objective functions $f_i:\mathbb{R}^{n_{xi}}\rightarrow \mathbb{R}$ and nonlinear inequality constraints $h_i: \mathbb{R}^{n_{xi}}\rightarrow \mathbb{R}^{n_{hi}}$ only depend on $x_i \in \mathbb{R}^{n_i}$ and that coupling between them takes place in the affine consensus constraint $\sum_{i\in \mathcal{R}}A_i x_i=0$ only.  There are several ways of formulating [<span style="font-variant:small-caps;">opf</span>]{}problems in form of differing in the coupling variables and the type of the power flow equations (polar or rectangular), cf. [ @Engelmann2019; @Erseghe2014a; @Kim1997]. 

**Initialization:** Initial guesses $z_i^0,\lambda_i^0$ for all $i \in \mathcal{R}$, parameter $\rho$.\
**Repeat** (until convergence):

1.   *Parallelizable Step:* Solve for all $i\in \mathcal{R}$ locally $$\begin{aligned}
    \label{eq:locStepADM}
            x_i^k=\underset{x_i}{\text{arg}\text{min}}&\quad f_i(x_i) + (\lambda^k_i)^\top A_i x_i + \frac{\rho}{2}\left\|A_i(x_i-z_i^k)\right\|_2^2\quad
            \text{s.t.}  \quad h_i(x_i) \leq 0.       
            \end{aligned}$$

2.   *Update dual variables* $$\label{eq:dualUp}
            \lambda_i^{k+1} = \lambda_i^{k} + \rho A_i(x_i^k-z_i^k)$$

3.   *Consensus Step:* Solve the coordination problem

    \[eq:ADMconsStep\] $$\begin{aligned}
                &\underset{\Delta x}{\text{min}}\;\sum_{i\in \mathcal{R}} \textstyle \frac{\rho}{2}\Delta x_i^\top A_i^\top A_i\Delta x_i + \lambda_i^{k+1\top}A_i \Delta x_i \\ 
                &\begin{aligned} \label{eq:ConsCnstrADM}
                \text{s.t.}                                 &\;\;\, \sum_{i\in \mathcal{R}}A_i(x^k_i+\Delta x_i) =  0     &&
                \end{aligned}
                \end{aligned}$$

4.   *Update variables* $\quad  \label{eq::update} 
            z^{k+1}\leftarrow  x^k + \Delta x^k. 
            $

\[alg:ADMM\]

**Initialization:** Initial guess $(z^0,\lambda^0)$, parameters $\Sigma_i\succ 0,\rho,\mu$.\
**Repeat** (until convergence):

1.   *Parallelizable Step:* Solve for all $i\in \mathcal{R}$ locally $$\begin{aligned}
    \notag
            x_i^k=\underset{x_i}{\text{arg}\text{min}}&\quad f_i(x_i) + (\lambda^k)^\top A_i x_i + \frac{\rho}{2}\left\|x_i-z_i^k\right\|_{\Sigma_i}^2\quad
            \text{s.t.}\quad h_i(x_i) \leq 0 \quad \mid \kappa_i^k.      
            \end{aligned}$$

2.   *Compute sensitivities:* Compute Hessian approximations $B_i^k$, gradients $g_i^k$ and Jacobians of the active constraints $C_i^k$, cf. [ @Houska2016]. 

3.   *Consensus Step:* Solve the coordination problem $$\begin{aligned}
            \notag
            &\underset{\Delta x,s}{\text{min}}\;\sum_{i\in \mathcal{R}}\left\{\textstyle \frac{1}{2}\Delta x_i^\top B^k_i\Delta x_i + {g_i^k}^\top \Delta x_i\right\} \hspace{-0.1em}  + \hspace{-0.1em} {\lambda^k}^\top  \hspace{-0.2em} s + \hspace{-0.1em}  \textstyle \frac{\mu}{2}\|s\|^2_2   \\ 
            &\begin{aligned} 
            \text{s.t.}                                 &\;\;\, \sum_{i\in \mathcal{R}}A_i(x^k_i+\Delta x_i) =  s     &&|\, \lambda^\text{QP}\\\label{eq::conqp} 
            &\;\;\;  C^k_i \Delta x_i = 0                                    &&\forall i\in \mathcal{R}. 
            \end{aligned}
            \end{aligned}$$

4.   *Update variables* $ \quad
            z^{k+1}\leftarrow  x^k + \Delta x^k,\;\;\; \lambda^{k+1} \leftarrow  \lambda^\mathrm{QP}. 
            $

    **Similarity to ADMM:** Remove $C_i^k$ in , set $B_i^k=\rho A_i^\top A_i, g_i^k=A_i^\top \lambda^k$ and $\Sigma_i=A_i^\top A_i$, set $\mu = \infty$ (i.e. $s=0$) and use dual ascent step for updating $\lambda^k$ in , cf. [ @Houska2016]. 

\[alg:ALADIN\]

Here, we are interested in solving Problem via [<span style="font-variant:small-caps;">admm</span>]{}and [<span style="font-variant:small-caps;">aladin</span>]{}summarized in Algorithm \[alg:ADMM\] and Algorithm \[alg:ALADIN\] receptively.[^2]$^,$[^3] At first glance it is apparent that [<span style="font-variant:small-caps;">admm</span>]{}and [<span style="font-variant:small-caps;">aladin</span>]{}share several features.  For example, in Step 1) of both algorithms, local augmented Lagrangians subject to local nonlinear inequality constraints $h_i$ are minimized in parallel.[^4] Observe that while [<span style="font-variant:small-caps;">admm</span>]{}maintains multiple local Lagrange multipliers $\lambda_i$, [<span style="font-variant:small-caps;">aladin</span>]{}considers one global Lagrange multiplier vector $\lambda$. In Step 2), [<span style="font-variant:small-caps;">aladin</span>]{}computes sensitivities $B_i$, $g_i$ and $C_i$ (which often can directly be obtained from the local numerical solver without additional computation) and [<span style="font-variant:small-caps;">admm</span>]{}updates the multiplier vectors $\lambda_i$.

In Step 3), both algorithms communicate certain information to a central entity which then solves a (usually centralized) coordination quadratic program.  However, [<span style="font-variant:small-caps;">aladin</span>]{}and [<span style="font-variant:small-caps;">admm</span>]{}differ in the amount of exchanged information: Whereas [<span style="font-variant:small-caps;">admm</span>]{}only communicates the the local primal [and dual]{} variables $x_i$ [and $\lambda_i$]{}, [<span style="font-variant:small-caps;">aladin</span>]{}additionally communicates sensitivities.  This is a considerable amount of extra information compared with [<span style="font-variant:small-caps;">admm</span>]{}.  However, there exist methods to reduce the amount of exchanged information and bounds on the information exchange are given in [@Engelmann2019].  Another important difference is the computational complexity of the coordination step.  In many cases, the coordination step in [<span style="font-variant:small-caps;">admm</span>]{}can be reduced to an averaging step based on neighborhood communication only [@Boyd2011], whereas in [<span style="font-variant:small-caps;">aladin</span>]{}the coordination step involves the centralized solution of an equality constrained quadratic program. 

In the last step, [<span style="font-variant:small-caps;">admm</span>]{}updates the primal variables $z_i$, while [<span style="font-variant:small-caps;">aladin</span>]{}additionally updates the dual variables $\lambda$. Differences of [<span style="font-variant:small-caps;">aladin</span>]{}and [<span style="font-variant:small-caps;">admm</span>]{}also show up in the convergence speed and their theoretical convergence guarantees: Whereas [<span style="font-variant:small-caps;">aladin</span>]{}guarantees global convergence and quadratic local convergence for non-convex problems if a certain line-search is applied [@Houska2016], few results exist for [<span style="font-variant:small-caps;">admm</span>]{}in the non-convex setting.  Recent works [@Wang2019; @Hong2016] investigate the convergence of [<span style="font-variant:small-caps;">admm</span>]{}for special classes of non-convex problems; however, to the best of our knowledge [<span style="font-variant:small-caps;">opf</span>]{}problems do not belong to these classes. 

Numerical Results {#sec:NumRes}
=================

Next, we investigate the behavior of [<span style="font-variant:small-caps;">admm</span>]{}for large $\rho$ and a feasible initialization to illustrate performance differences between [<span style="font-variant:small-caps;">admm</span>]{}and [<span style="font-variant:small-caps;">aladin</span>]{}.  We consider power flow equations in polar form with coupling in active/reactive power and voltage angle and magnitude at the boundary between two neighbored regions [@Engelmann2019].  We consider the [<span style="font-variant:small-caps;">ieee</span>]{}57-bus system with data from <span style="font-variant:small-caps;">matpower</span> and partitioning as in [@Engelmann2019] as numerical test case. 

Figures \[fig:ADMAL1e4\] and \[fig:ADMAL1e6\] show the convergence behavior of [<span style="font-variant:small-caps;">admm</span>]{}(with and without feasible initialization (f.)) and [<span style="font-variant:small-caps;">aladin</span>]{}for several convergence indicators and two different penalty parameters $\rho = 10^4$ and $\rho = 10^6$.[^5] Therein, the left-handed plot depicts the consensus gap $\|Ax^k\|_\infty$ representing the maximum mismatch of coupling variables (active/reactive power and voltage magnitude/angle) at borders between two neighbored regions.  The second plot shows the objective function value $f^k$ and the third plot presents the distance to the minimizer $\|x^k-x^\star\|_\infty$ over the iteration index $k$. The right-handed figure shows the nonlinear constraint violation $\|g(z^k)\|_\infty$ after the consensus steps and of Algorithm \[alg:ADMM\] and \[alg:ALADIN\] respectively representing the maximum violation of the power flow equations.[^6]

\[fig:ADMAL1e4\]

In case of small $\rho = 10^4$, both [<span style="font-variant:small-caps;">admm</span>]{}variants behave similar and converge slowly towards the optimal solution with slow decrease in consensus violation, nonlinear constraint violation and objective function value.  If we increase $\rho$ to $\rho=10^{6}$ with results shown in Figure \[fig:ADMAL1e6\], the consensus violation $\|Ax^k\|_\infty$ gets smaller in [<span style="font-variant:small-caps;">admm</span>]{}with feasible initialization.  The reason is that a large $\rho$ forces $x_i^k$ being close to $z_i^k$ leading to small $\|Ax^k\|_\infty$ as we have $Az^k=0$ from the consensus step.  But, at the same time, this also leads to a slower progress in optimality $f^k$ compared to $\rho =10^4$, cf.  the second plot in Figures \[fig:ADMAL1e4\] and \[fig:ADMAL1e6\]. 

[On the other hand, this statement does not hold for [<span style="font-variant:small-caps;">admm</span>]{}with infeasible initialization (blue lines in Figures \[fig:ADMAL1e4\] and \[fig:ADMAL1e6\]) as the constraints in the local step and the consensus step of [<span style="font-variant:small-caps;">admm</span>]{}enforce an alternating projection between the consensus constraint and the local nonlinear constraints.  The progress in the nonlinear constraint violation $\|g(z^k)\|_\infty$ supports this statement.  In its extreme, this behavior can be observed when using $\rho = 10^{12}$ depicted in Figure \[fig:ADMAL1e12\].  There, [<span style="font-variant:small-caps;">admm</span>]{}with feasible initialization produces very small consensus violations and nonlinear constraint violations at cost of almost no progress in terms of optimality. ]{} 

Here the crucial observation is that in case of feasible initialization and large penalization parameter $\rho$ [<span style="font-variant:small-caps;">admm</span>]{}produces *almost feasible* iterates at cost of slow progress in the objective function values.  From this, one is tempted to conclude that also for infeasible initializations [<span style="font-variant:small-caps;">admm</span>]{}will likely converge, cf.  Figure \[fig:ADMAL1e4\]-\[fig:ADMAL1e12\].  This conclusion is supported by the rather small 57-bus test system.  However, it deserves to be noted that this conclusion is in general not valid, cf. [ @Guo2017]. 

For [<span style="font-variant:small-caps;">aladin</span>]{}, we use $\rho = 10^6$ and $\mu = 10^7$. Comparing the results of [<span style="font-variant:small-caps;">admm</span>]{}with [<span style="font-variant:small-caps;">aladin</span>]{}, [<span style="font-variant:small-caps;">aladin</span>]{}shows superior quadratic convergence also in case of infeasible initialization.  This is inline with the known convergence properties of [<span style="font-variant:small-caps;">aladin</span>]{}[@Houska2016].  However, the fast convergence comes at the cost of increased communication overhead per step, cf. [ @Engelmann2019] for a more detailed discussion.  Note that [<span style="font-variant:small-caps;">aladin</span>]{}involves a more complex coordination step, which is not straightforward to solve via neighborhood communication.  Furthermore, tuning of $\rho$ and $\mu$ can be difficult. 

In power systems, usually feasibility is preferred over optimality to ensure a stable system operation.  Hence, [<span style="font-variant:small-caps;">admm</span>]{}with feasible initialization and large $\rho$ can in principle be used for [<span style="font-variant:small-caps;">opf</span>]{}as in this case violation of power flow equations and generator bounds are expected to be small and one could at least expect certain progress towards an optimal solution.  Following this reasoning several papers consider $\|A(x^k-z^k)\|_\infty <\epsilon$ as termination criterion [@Erseghe2014a; @Guo2017].  However, as shown in the example above, if $\rho$ is large enough, and [<span style="font-variant:small-caps;">admm</span>]{}is initialized at a feasible point, this termination criterion can always be satisfied in just one iteration if $\rho$ is chosen sufficiently large.  Consequently, it is unclear how to ensure a certain degree of optimality.  An additional question with respect to [<span style="font-variant:small-caps;">admm</span>]{}is how to obtain a feasible initialization.  To compute such an initial guess, one has to solve a constrained nonlinear least squares problem solving the power flow equations subject to box constraints.  This is itself a problem of almost the same complexity as the full [<span style="font-variant:small-caps;">opf</span>]{}problem.  Hence one would again require a computationally powerful central entity with full topology and parameter information.  Arguably this jeopardizes the initial motivation for using distributed optimization methods. 

Analysis of ADMM with feasible initialization for $\rho \rightarrow \infty$ {#sec:Theory}
===========================================================================

The results above indicate that large penalization parameters $\rho$ in combination with feasible initialization might lead to pre-mature convergence to a suboptimal solution.  Arguably, this behavior of [<span style="font-variant:small-caps;">admm</span>]{}might be straight-forward to see from an optimization perspective.  However, to the best of our knowledge a mathematically rigorous analysis, which is very relevant for [<span style="font-variant:small-caps;">opf</span>]{}, is not available in the literature. 

\[lem:ADMMfeas\]\
Consider the application of [<span style="font-variant:small-caps;">admm</span>]{}(Algorithm \[alg:ADMM\]) to Problem .  Suppose that, for all $k \in \mathbb{N}$, the local problems have unique regular minimizers $x_i^k$.[^7] For $\tilde k\in \mathbb{N}$, let $\lambda^{\tilde k}_i$ be bounded and, for all $i\in \mathcal{R}$, $z_i^{\tilde k} \in \mathcal{X}_i$. Then, the [<span style="font-variant:small-caps;">admm</span>]{}iterates satisfy $$\lim_{\rho \to \infty}x_i^{k}(\rho) - x_i^{\tilde k} \in \operatorname{null}(A_i),\quad \forall k>\tilde k.$$

The proof is divided into four steps.  Steps 1)-3) establish technical properties used to derive the above assertion in Step 4). 

At iteration $\tilde k$ the local steps of [<span style="font-variant:small-caps;">admm</span>]{}are $$\begin{aligned}
\label{eq:ProofMin}
        x_i^{\tilde k}(\rho) = \underset{x_i\in \mathcal{X}_i}{\operatorname{argmin}}& \;f_i(x_i) + \left(\lambda_i^{\tilde k}\right)^\top A_i x_i + \frac{\rho}{2}\left\|A_i\left(x_i-z_i^{\tilde k}\right)\right\|^2_2.         \end{aligned}$$ Now, by assumption all $f_i$ is twice continuously differentiable (hence bounded on $\mathcal{X}_i$), $\lambda_i^{\tilde k}$ is bounded and all ${z_i^{\tilde k}\in \mathcal{X}_i}$. Thus, for all $i \in \mathcal{R}$, $\underset{\rho \rightarrow \infty}{\lim} x_i^{\tilde k}(\rho) = z_i^{\tilde k} + v_i^{\tilde k}$ with $v_i^{\tilde k} \in \operatorname{null}(A_i)$.

The first-order stationarity condition of can be written as $$\begin{aligned}
\label{eq:statCond}
        -\nabla f_i(x_i^{\tilde k}) - \gamma_i ^{\tilde k\top} \nabla h_i( x_i^{\tilde k}) = A_i^\top \lambda_i^{\tilde k} + \rho A_i^\top A_i\left(x_i^{\tilde k}-z_i^{\tilde k}\right),
    \end{aligned}$$ where $\gamma_i ^{\tilde k\top}$ is the multiplier associated to $h_i$. Multiplying the multiplier update formula with $A_i^\top$ from the left we obtain $A_i^\top\lambda_i^{k+1} = A_i^\top\lambda_i^{k} + \rho A_i^\top A_i(x_i^k-z_i^k)$. Combined with this yields $       A_i^\top \lambda_i^{\tilde k+1} = - \nabla f(x_i^{\tilde k}) - \gamma ^{\tilde k\top} \nabla h_i(x_i^{\tilde k})$. By differentiability of $f_i$ and $h_i$, compactness of $\mathcal{X}_i$ and regularity of $x_i^{\tilde k}$ this implies boundedness of $A_i^\top \lambda_i^{\tilde k+1}$.

Next, we show by contradiction that $\Delta x_i^{\tilde k} \in \operatorname{null}(A_i)$ for all $i \in \mathcal{R}$ and $\rho \rightarrow \infty$. Recall the coordination step in [<span style="font-variant:small-caps;">admm</span>]{}given by $$\begin{aligned}
 \label{eq:proofQP}
            \underset{\Delta x}{\text{min}}\;\sum_{i\in \mathcal{R}} \dfrac{\rho}{2}\Delta x_i^\top A_i^\top A_i\Delta x_i + \lambda_i^{\tilde k+1\top}A_i \Delta x_i 
            \;\;\;\text{s.t.} \;                                \; \sum_{i\in \mathcal{R}}A_i(x^{\tilde k}_i+\Delta x_i) =  0.  
            \end{aligned}$$ Observe that any $\Delta x_i^{\tilde k} \in \operatorname{null}(A_i)$ is a feasible point to as $\sum_{i\in \mathcal{R}}A_ix^{\tilde k}_i=0$. Consider a feasible candidate solution $\Delta x_i \notin \operatorname{null}(A_i)$ for which $ \sum_{i\in \mathcal{R}}A_i(x^{\tilde k}_i+\Delta x_i) =  0$. Clearly, $\lambda_i^{\tilde k+1\top}A_i \Delta x_i(\rho)$ will be bounded.  Hence for a sufficiently large value of $\rho$, the objective of will be positive.  However, for any $\Delta x_i  \in \operatorname{null}(A_i)$ the objective of is zero, which contradicts optimality of the candidate solution $\Delta x_i \notin \operatorname{null}(A_i)$. Hence, choosing $\rho$ sufficiently large ensures that any minimizer of lies in $\operatorname{null}(A_i)$.

It remains to show $x_i^{\tilde k + 1} = x_i^{\tilde k}$. In the last step of [<span style="font-variant:small-caps;">admm</span>]{}we have $z^{\tilde k+1}=x^{\tilde k} + \Delta x^{\tilde k}$. Given Steps 1-3) this yields $z^{\tilde k+1}=z^{\tilde k} + v^{\tilde k} + \Delta x^{\tilde k}$ and hence $$\left\|A_i\left(x_i-z_i^{ \tilde k+1}\right)\right\|^2_2=\left\|A_i\left(x_i-z_i^{\tilde k} + v_i^{\tilde k} + \Delta x_i^{\tilde k}\right)\right\|^2_2=\left\|A_i\left(x_i-z_i^{\tilde k}\right)\right\|^2_2.$$ Observe that this implies that, for $\rho \rightarrow \infty$, problem $\eqref{eq:ProofMin}$ does not change from step $\tilde k$ to $\tilde k+1$. This proves the assertion.  $\hfill \blacksquare$

\
Assuming that the local subproblems in [<span style="font-variant:small-caps;">admm</span>]{}are solved deterministically; i.e. same problem data yields the same solution.  Then under the conditions of Proposition \[lem:ADMMfeas\] and for $\rho \rightarrow \infty$, once [<span style="font-variant:small-caps;">admm</span>]{}generates a feasible point $x_i^{\tilde k}$ to Problem , or whenever it is initialized at a feasible point, it will stay at this point for all subsequent $k>\tilde k$.

The above corollary explains the behavior of [<span style="font-variant:small-caps;">admm</span>]{}for large $\rho$ in combination with feasible initialization often used in power systems [@Guo2017; @Erseghe2014a].  Despite feasible iterates are desirable from a power systems point of view, the findings above imply that high values of $\rho$ limit progress in terms of minimizing the objective. 

\
Note that for $\rho \to \infty$, [<span style="font-variant:small-caps;">aladin</span>]{}behaves different than [<span style="font-variant:small-caps;">admm</span>]{}.  While the local problems in [<span style="font-variant:small-caps;">aladin</span>]{}behave similar to [<span style="font-variant:small-caps;">admm</span>]{}, the coordination step in [<span style="font-variant:small-caps;">aladin</span>]{}is equivalent to a sequential quadratic programming step.  This helps avoiding premature convergence and it ensures decrease of $f$ in the coordination step [@Houska2016]. 

Conclusions
===========

This method-oriented work investigated the interplay of penalization of consensus violation and feasible initialization in [<span style="font-variant:small-caps;">admm</span>]{}.  We found that—despite often working reasonably with a *good choice* of $\rho$ and infeasible initialization—in case of feasible initialization combined with large values of $\rho$ [<span style="font-variant:small-caps;">admm</span>]{}typically stays feasible yet it may stall at a suboptimal solution.  We provided analytical results supporting this observation.  However, computing a feasible initialization is itself a problem of almost the same complexity as the full [<span style="font-variant:small-caps;">opf</span>]{}problem; in some sense partially jeopardizing the advantages of distributed optimization methods.  Thus distributed methods providing rigorous convergence guarantees while allowing for infeasible initialization are of interest.  One such alternative method is [<span style="font-variant:small-caps;">aladin</span>]{}[@Houska2016] exhibiting convergence properties at cost of an enlarged communication overhead and a more complex coordination step [@Engelmann2019]. 

[^1]: [This work is part of a project that receives funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No.  730936.  TF acknowledges further support from the Baden-Württemberg Stiftung under the Elite Programme for Postdocs.]{} 

[^2]: [We remark that there exist a variety of variants of [<span style="font-variant:small-caps;">admm</span>]{}, cf. [ @Boyd2011; @Bertsekas1989].  Here, we choose the formulation from [@Houska2016] in order to highlight similarities between [<span style="font-variant:small-caps;">admm</span>]{}and [<span style="font-variant:small-caps;">aladin</span>]{}.]{} 

[^3]: Note that, due to space limitations, we describe the full-step variant of [<span style="font-variant:small-caps;">aladin</span>]{}here.  To obtain convergence guarantees from a remote starting point, a line-search is necessary, cf. [ @Houska2016]. 

[^4]: For notational simplicity, we only consider nonlinear inequality constraints here.  Nonlinear equality constraints $g_i$ can be incorporated via a reformulation in terms of two inequality constraints, i.e. $0\leq g_i(x_i) \leq 0$.

[^5]: The scaling matrices $\Sigma_i$ are diagonal.  They scale to improve convergence.  Hence, entries corresponding to voltages and phase angles are $100$, entries corresponding to powers are $1$.

[^6]: The power flow equations for the [<span style="font-variant:small-caps;">ieee</span>]{}57-bus systems are considered as nonlinear equality constraints $g_i(x_i)=0$. Hence, $g_i(x_i)\neq0$ represents a violation of the power flow equations. 

[^7]: A minimizer is regular if the gradients of the active constraints are linear independent [@Nocedal2006].