Skip to main content

Advertisement

Table 1 Comparison of ADMM update rules at node 4 using C-CADMM, D-CADMM, and H-CADMM to solve the problem in Example 1

From: Hybrid ADMM: a unifying and fast approach to decentralized optimization

Method Update rules
C-CADMM \({x}_{4}^{k+1} = \underset {{x}_{4}}{\text {arg\,min}} \left [ f_{4}({x}_{4}) + {\lambda }_{4}^{k}\left ({x}_{4} - \bar {{x}}^{k}\right) + \frac {{\rho }}{2}\|{x}_{4} - \bar {{x}}^{k}\|_{2}^{2}\right ] \)
  \( {\lambda }_{4}^{k+1} = {\lambda }_{4}^{k} + {\rho }{}\left ({x}_{4}^{k+1} - \bar {{x}}^{k+1}\right)\)
D-CADMM \({x}_{4}^{k+1} = (\nabla f_{4} + 2{\rho } I)^{-1} \left ({\rho } {x}_{4}^{k} + \frac {{\rho }}{2}\left ({x}_{2}^{k} + {x}_{5}^{k}\right) - {y}_{4}^{k}\right) \)
  \({y}_{4}^{k+1} = {y}_{4}^{k} + \frac {{\rho }}{2}\left (2{x}_{4}^{k+1} - {x}_{2}^{k+1} - {x}_{5}^{k+1}\right) \)
H-CADMM \({x}_{4}^{k+1} = \underset {{x}_{4}}{\text {arg\,min}} \Bigg [ f_{4}({x}_{4}) + {y}_{4}^{k} {x}_{4} + \frac {{\rho }}{2} \Big (\|{x}_{4} - \frac {1}{4}\sum _{i=1}^{4} {x}_{i}^{k}\|^{2} + \|{x}_{4} - \frac {1}{2} \sum _{i=1}^{2} {x}_{i}^{k}\|^{2} \Big)\Bigg ] \)
  \({y}_{4}^{k+1} = {y}_{4}^{k} + {\rho }\left (2{x}_{4}^{k+1} - \frac {1}{4}\sum _{i=1}^{4} {x}_{i}^{k+1} - \frac {1}{2} \sum _{i=4}^{5} {x}_{i}^{k+1}\right) \)