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where the dual variable is chosen such that the power constraint in (4) is satisfied with equality.

A valid supergradient for each particular dual variable is given by the constraint it is associated with [31].

The dual variable ((sigma)) is obtained from the fourth constraint of Model (3).

In this case the core of the algorithm is the primal iteration (S1) and the dual variable λ(t) is an internal state that determines the operating point.

Two dual variables are introduced, which are λ bw and λ rate.

where η is the diminishing step size as the update of dual variable is performed according to the diminishing step approach [26] for each user k.

The local variable and dual variable updates are carried out independently in parallel by all BSs, while the global variable update is carried out by BS coordination.

New iterative methods for the solution of the unsymmetrical dual variable system are developed and are proven to converge for a large class of problems.

Then, according to ADMM, the dual variable can be updated by left{ begin{aligned} &mathbf{v}_{n}left({q + 1} right) = {mathbf{v}_{n}}left(q right) + cleft {boldsymbol{Omega}_{n}mathbf{t}left({q + 1}right) - {mathbf{t}_{n}}left({q + 1} right)} right), &{u_{n}}left({q + 1} right) = {u_{n}}left(q right) + cleft({{rho_{n}}left({q + 1} right) - {p_{n}}left({q +1} right)} right) end{aligned} right.

According to De Serpa, in activities, such as leisure activities, where constraint does not come into play (when an individual devotes more time than the minimum required for the consumption of a good), the dual variable will be zero and therefore the value of time saving in the consumption of the activity will also be zero.

Determining optimal dual variables Λ∗ is easier than determining optimal primal pairs (p∗,x∗) because there are a finite number of multipliers and the dual function is convex.