The energy efficiency optimal algorithm is given by the following subgradient
Duchi, John, Adaptive Journal of Machine EladHazan, and subgradient
Learning Research Yoram Singer.
In , the authors proposed a subgradient
In the developed algorithm, the scheduling indicator is evaluated by the central controller and represented in a function of [[bar.[lambda]].sup.(t)] where [[bar.[lambda]].sup.(t)] stands for the Lagrangian multiplier vector of the dual problem in (2), which is represented after the transmission as a function of [[alpha].sup.(t)], a step size at time-slot t, and [V.sub.k.sub.(t)] the stochastic subgradient
of the dual problem in (1) which can be obtained by Danskin's Theorem  as
Now, we minimize the dual problem using subgradient
The inexact SA algorithm solves (8) by mimicking the subgradient
Eventually, a subgradient
search is adopted to iteratively update the feasible and relaxed solutions to an acceptable tolerance of optimality gap.
At the first stage, the assignment of berths and QCs to vessels is determined by Lagrangian relaxation and a subgradient
They used the modified subgradient
algorithm to solve this non-linear programming problem.
To minimize J([[theta].sub.0]), we use a generalization of gradient descent called subgradient
method  which computes a gradient-like direction.