| $\beta$ |
hyperparameter for subgradient method |
$[1;\infty[$ |
| $\Box$ |
generalized controller optimization variable |
$---$ |
| $c$ |
coupling constraint function |
$\mathbb{R}^{n_c}$ |
| $c_h$ |
coupling constraint component (mapping) |
$\mathbb{R}^{n_h}$ |
| $c_{h}$ |
coupling constraint component (mapping) |
$\mathbb{R}^{n_h}$ |
| $c_z$ |
coupling constraint component (shared variables) |
$\mathbb{R}^{n_z}$ |
| $c_{z}$ |
coupling constraint component (shared variables) |
$\mathbb{R}^{n_z}$ |
| $c_c$ |
consensus constraint function |
$\mathbb{R}^{n_y}$ |
| $d$ |
design variable |
$\mathbb{R}^{n_d} \times \mathbb{C}^{n_d}$ |
| $d^{*}$ |
optimal design variable |
$\mathbb{R}^{n_d} \times \mathbb{C}^{n_d}$ |
| $d_{1}$ |
first component of design variable |
$\mathbb{R}$ |
| $d_{n_d}$ |
last component of design variable |
$\mathbb{R}$ |
| $D$ |
design variable domain (non-calligraphic notation) |
$---$ |
| $\mathcal{D}$ |
set of design variable |
$\mathbb{R}^{n_d} \times \mathbb{C}^{n_d}$ |
| $\epsilon_k$ |
tolerance hyperparameter for SBDP outerloop convergence criterion |
$\mathbb{R}^{+}$ |
| $\gamma$ |
hyperparameter for subgradient method |
$]0;1[$ |
| $h$ |
coupling variable |
$\mathbb{R}^{n_h} \times \mathbb{C}^{n_h}$ |
| $h^{*}$ |
optimal coupling variable |
$\mathbb{R}^{n_h} \times \mathbb{C}^{n_h}$ |
| $\mathcal{H}$ |
set of additional coupling design variable |
$\mathbb{R}^{n_h} \times \mathbb{C}^{n_h}$ |
| $H$ |
mapping operator |
$-$ |
| $i$ |
subsystem index |
$i \in M$ |
| $j$ |
neighboring subsystem index |
$j \in N$ |
| $k$ |
outerloop iterator |
$\mathbb{I}_0^{+}$ |
| $\kappa_d$ |
Lagrange multiplier associated with active bounds of design variable set D |
$\mathbb{R}^{...}$ |
| $\kappa_g$ |
Lagrange multiplier associated with local inequality constraint v_g |
$\mathbb{R}^{n_{v_g}}$ |
| $\kappa_h$ |
Lagrange multiplier associated with local equality constraint v_h |
$\mathbb{R}^{n_{v_h}}$ |
| $l$ |
innerloop iterator |
$\mathbb{I}_0^{+}$ |
| $L$ |
(augmented) Lagrange function |
$\mathbb{R}$ |
| $\lambda$ |
Lagrange multiplier associated with $c$ or $c_c$ |
$\mathbb{R}^{n_c}$ |
| $\lambda_h$ |
coordination-equality multiplier (mapped-response block) |
$\mathbb{R}^{n_h}$ |
| $\lambda_z$ |
coordination-equality multiplier (shared-design-variable block) |
$\mathbb{R}^{n_z}$ |
| $m$ |
index of last subsystem |
$M$ |
| $M$ |
set of indices of subsystems |
$\mathbb{I}_0^{+}$ |
| $N$ |
set of indices of neighboring subsystems |
$\mathbb{I}_0^{+}$ |
| $n_c$ |
dimension of coupling constraint function |
$\mathbb{I}^{+}$ |
| $n_d$ |
dimension of design variable |
$\mathbb{I}^{+}$ |
| $n_h$ |
dimension of coupling variable |
$\mathbb{I}^{+}$ |
| $n_x$ |
dimension of local design variable |
$\mathbb{I}^{+}$ |
| $n_y$ |
dimension of auxiliary variable |
$\mathbb{I}^{+}$ |
| $n_z$ |
dimension of shared design variable |
$\mathbb{I}^{+}$ |
| $n_{v_g}$ |
dimension of local inequality constraint function |
$\mathbb{I}^{+}$ |
| $n_{v_h}$ |
dimension of local equality constraint function |
$\mathbb{I}^{+}$ |
| $\nu$ |
penalty parameter |
$\mathbb{R}_+$ |
| $p$ |
coordination parameters in unified notation |
$...$ |
| $P$ |
objective term for coordination |
$...$ |
| $Q$ |
constraint term for coordination |
$...$ |
| $Q^{\leq}$ |
inequality constraint term for coordination |
$...$ |
| $Q^{=}$ |
equality constraint term for coordination |
$...$ |
| $r$ |
local response function |
$\mathbb{R}$ |
| $s$ |
penalty parameter |
$\mathbb{R}^{n_c}$ |
| $S$ |
selector matrix to retrieve ${}^ix,\:{}^i_jz,\:{}^i_jh$ from ${}^id$ |
$---$ |
| $S_z$ |
selector matrix for shared design variables |
$---$ |
| $S_{z}$ |
selector matrix for shared design variables |
$---$ |
| $S_{h}$ |
selector matrix for coupling variables |
$---$ |
| $\Sigma$ |
positive definite scaling matrix |
$\mathbb{R}^{n_d}$ |
| $\tau$ |
slack variable |
$\mathbb{R}^{n_c}$ |
| $u$ |
coupling information in unified notation |
$...$ |
| $v_f$ |
local objective function |
$\mathbb{R}$ |
| $v_g$ |
local inequality constraint function |
$\mathbb{R}^{n_g}$ |
| $v_h$ |
local equality constraint function |
$\mathbb{R}^{n_h}$ |
| $v_D$ |
local box constraint function on design variables |
$---$ |
| $v_{\mathcal{D}}$ |
local box inequality constraint function |
$---$ |
| $x$ |
local design variable |
$\mathbb{R}^{n_x} \times \mathbb{C}^{n_x}$ |
| $x^{*}$ |
optimal local design variable |
$\mathbb{R}^{n_x} \times \mathbb{C}^{n_x}$ |
| $\mathcal{X}$ |
set of local design variable |
$\mathbb{R}^{n_x} \times \mathbb{C}^{n_x}$ |
| $y$ |
auxiliary variable |
$\mathbb{R}^{n_y}$ |
| $z$ |
shared design variable |
$\mathbb{R}^{n_z} \times \mathbb{C}^{n_z}$ |
| $z^{*}$ |
optimal shared design variable |
$\mathbb{R}^{n_z} \times \mathbb{C}^{n_z}$ |
| $z_{\{i,j\}}$ |
condensed shared design variable between subsystems $i$ and $j$ |
$\mathbb{R}^{n_z} \times \mathbb{C}^{n_z}$ |
| $z_{\{i,j\}}^{*}$ |
optimal condensed shared design variable |
$\mathbb{R}^{n_z} \times \mathbb{C}^{n_z}$ |
| $z_{\{0,j\}}$ |
condensed shared design variable between subsystem 0 and $j$ |
$\mathbb{R}^{n_z} \times \mathbb{C}^{n_z}$ |
| $z_{\{m,j\}}$ |
condensed shared design variable between subsystem $m$ and $j$ |
$\mathbb{R}^{n_z} \times \mathbb{C}^{n_z}$ |
| $z_{t}$ |
target shared design variable |
$\mathbb{R}^{n_z} \times \mathbb{C}^{n_z}$ |
| $\mathcal{Z}$ |
set of shared design variable |
$\mathbb{R}^{n_z} \times \mathbb{C}^{n_z}$ |