Unified Algorithmic Structure

Although the previously introduced methods — ALC, ALADIN, SBDP, and Consensus ALC — differ in methodology, they share a unified structure. Each subsystem solves

\[\begin{aligned} {}^{i}d^{(k,\,l+1)} \leftarrow{} & \argmin_{{}^{i}d\;\in\;{}^{i}\mathcal{D}} \;\; {}^{i}v_f\!\left({}^{i}r\right) + {}^{i}P\!\left({}^{i}d,\; \left\{{}^{i}_{j}u,\;{}^{j}_{i}u\right\}_{j\in{}^{i}N},\; {}^{i}_{C}u,\;{}^{C}_{i}u\right) \\ & \text{s.t.} \;\; {}^{i}v_g\!\left({}^{i}r\right) \leq 0, \\ & \phantom{\text{s.t.}} \;\; {}^{i}v_h\!\left({}^{i}r\right) = 0, \\ & \phantom{\text{s.t.}} \;\; {}^{i}Q^{\leq}\!\left({}^{i}d,\; \left\{{}^{i}_{j}u,\;{}^{j}_{i}u\right\}_{j\in{}^{i}N},\; {}^{i}_{C}u,\;{}^{C}_{i}u\right) \leq 0, \\ & \phantom{\text{s.t.}} \;\; {}^{i}Q^{=}\!\left({}^{i}d,\; \left\{{}^{i}_{j}u,\;{}^{j}_{i}u\right\}_{j\in{}^{i}N},\; {}^{i}_{C}u,\;{}^{C}_{i}u\right) = 0, \end{aligned}\]

where \({}^{i}\)\(v_f\), \({}^{i}\)\(v_g\), and \({}^{i}\)\(v_h\) are private to subsystem \(i\). The coordination approach may introduce an objective term \({}^{i}\)\(P\) or constraint term \({}^{i}\)\(Q\) that depend on coupling information \(u\) exchanged with neighbors and, optionally, a controller. The controller solves

\[\begin{aligned} \Box \leftarrow{} & \argmin_{\Box} \;\; {}^{C}P\!\left(\Box,\; \left\{{}^{C}_{i}u,\;{}^{i}_{C}u\right\}_{i\in M}\right) \\ & \text{s.t.} \;\; {}^{C}Q^{\leq}\!\left(\Box,\; \left\{{}^{C}_{i}u,\;{}^{i}_{C}u\right\}_{i\in M}\right) \leq 0, \\ & \phantom{\text{s.t.}} \;\; {}^{C}Q^{=}\!\left(\Box,\; \left\{{}^{C}_{i}u,\;{}^{i}_{C}u\right\}_{i\in M}\right) = 0. \end{aligned}\]

The unified algorithm alternates between inner and outer loops—solving subsystem and/or controller problems, updating coupling parameters (e.g. multipliers or penalty parameters), and exchanging \(u\) through storage interfaces. Not every method uses all parts:

• the controller problem may be absent,
• the inner loop may collapse to a single iteration or be omitted entirely,
• coupling parameters may be updated in the inner loop, the outer loop, or both.

Algorithm 5 therefore represents a generalized superset of the previously derived methods.

All symbols are defined in the Nomenclature.

Algorithm 5 Unified Algorithmic Structure

Require: hyperparameters for inner and outerloop convergence criteria

Require: hyperparameters for update of relevant coupling parameters

Require: initial relevant coupling parameters in interface storage

 

\(k \leftarrow 0\) ▷ initialize outerloop iterator

repeat

\(l \leftarrow 0\) ▷ initialize innerloop iterator

repeat ▷ following some iteration scheme

for every \(i \in M\) do

Copy relevant coupling parameters of \({}^{i}_{j}u,\; {}^{j}_{i}u \;\forall\; j \in {}^{i}N\) and \({}^{i}_{C}u,\; {}^{C}_{i}u\) from interface storage

Prepare optimization problem formulation

\[\begin{aligned} {}^{i}d^{(k,\,l+1)} \leftarrow{} & \argmin_{{}^{i}d \;\in\; {}^{i}\mathcal{D}} \;\; {}^{i}v_f\!\left({}^{i}r\right) + {}^{i}P\!\left({}^{i}d,\; \left\{{}^{i}_{j}u,\; {}^{j}_{i}u\right\}_{j \in {}^{i}N},\; {}^{i}_{C}u,\; {}^{C}_{i}u\right) \\ & \text{s.t.} \;\; {}^{i}v_g\!\left({}^{i}r\right) \leq 0, \\ & \phantom{\text{s.t.}} \;\; {}^{i}v_h\!\left({}^{i}r\right) = 0, \\ & \phantom{\text{s.t.}} \;\; {}^{i}Q^{\leq}\!\left({}^{i}d,\; \left\{{}^{i}_{j}u,\; {}^{j}_{i}u\right\}_{j \in {}^{i}N},\; {}^{i}_{C}u,\; {}^{C}_{i}u\right) \leq 0, \\ & \phantom{\text{s.t.}} \;\; {}^{i}Q^{=}\!\left({}^{i}d,\; \left\{{}^{i}_{j}u,\; {}^{j}_{i}u\right\}_{j \in {}^{i}N},\; {}^{i}_{C}u,\; {}^{C}_{i}u\right) = 0. \end{aligned}\]

Post-process the optimization

Copy relevant coupling parameters of \({}^{i}_{j}u,\; {}^{j}_{i}u \;\forall\; j \in {}^{i}N\) and \({}^{i}_{C}u,\; {}^{C}_{i}u\) to interface storage

end for

 

for Controller do

Copy relevant coupling parameters of \({}^{C}_{i}u,\; {}^{i}_{C}u \;\forall\; i \in M\) from interface storage

Prepare optimization problem formulation

\[\begin{aligned} \Box \leftarrow{} & \argmin_{\Box} \;\; {}^{C}P\!\left(\Box,\; \left\{{}^{C}_{i}u,\; {}^{i}_{C}u\right\}_{i \in M}\right) \\ & \text{s.t.} \;\; {}^{C}Q^{\leq}\!\left(\Box,\; \left\{{}^{C}_{i}u,\; {}^{i}_{C}u\right\}_{i \in M}\right) \leq 0, \\ & \phantom{\text{s.t.}} \;\; {}^{C}Q^{=}\!\left(\Box,\; \left\{{}^{C}_{i}u,\; {}^{i}_{C}u\right\}_{i \in M}\right) = 0. \end{aligned}\]

Post-process the optimization

Copy relevant coupling parameters of \({}^{C}_{i}u,\; {}^{i}_{C}u \;\forall\; i \in M\) to interface storage

end for

 

for every \(i \in M\) and Controller in parallel do

Copy relevant coupling parameters of \({}^{i}_{j}u,\; {}^{j}_{i}u \;\forall\; j \in {}^{i}N\) and \({}^{i}_{C}u,\; {}^{C}_{i}u\) from interface storage

Update relevant coupling parameters \({}^{i}_{j}u,\; {}^{j}_{i}u \;\forall\; j \in {}^{i}N\) and \({}^{i}_{C}u,\; {}^{C}_{i}u\)

Copy relevant coupling parameters of \({}^{i}_{j}u,\; {}^{j}_{i}u \;\forall\; j \in {}^{i}N\) and \({}^{i}_{C}u,\; {}^{C}_{i}u\) to interface storage

end for

 

Compute innerloop convergence criterion

\(l \leftarrow l + 1\)

until innerloop convergence criterion is met

 

\({}^{i}d^{(k+1)} \leftarrow {}^{i}d^{(k,\,l)} \;\forall\; i \in M\)

 

for every \(i \in M\) and Controller in parallel do

Copy relevant coupling parameters of \({}^{i}_{j}u,\; {}^{j}_{i}u \;\forall\; j \in {}^{i}N\) and \({}^{i}_{C}u,\; {}^{C}_{i}u\) from interface storage

Prepare update of relevant coupling parameters \({}^{i}_{j}u,\; {}^{j}_{i}u \;\forall\; j \in {}^{i}N\) and \({}^{i}_{C}u,\; {}^{C}_{i}u\)

Copy relevant coupling parameters of \({}^{i}_{j}u,\; {}^{j}_{i}u \;\forall\; j \in {}^{i}N\) and \({}^{i}_{C}u,\; {}^{C}_{i}u\) to interface storage

end for

 

for every \(i \in M\) and Controller in parallel do

Copy relevant coupling parameters of \({}^{i}_{j}u,\; {}^{j}_{i}u \;\forall\; j \in {}^{i}N\) and \({}^{i}_{C}u,\; {}^{C}_{i}u\) from interface storage

Update relevant coupling parameters \({}^{i}_{j}u,\; {}^{j}_{i}u \;\forall\; j \in {}^{i}N\) and \({}^{i}_{C}u,\; {}^{C}_{i}u\)

Copy relevant coupling parameters of \({}^{i}_{j}u,\; {}^{j}_{i}u \;\forall\; j \in {}^{i}N\) and \({}^{i}_{C}u,\; {}^{C}_{i}u\) to interface storage

end for

 

Compute outerloop convergence criterion

\(k \leftarrow k + 1\)

until outerloop convergence criterion is met

return \(\left\{{}^{i}d^{(k)}\right\}_{i \in M}\)