## 5 Properties Analysis of the Functions \(\varphi_{m}^{s}\)
### Single-Resource-Block Maritime User Allocation
When calculating the optimal \(W^{s}_{RB}\) corresponding to \(Q\) different power budgets in a multi-resource-block problem, the RBPUA with a computational complexity of \(O(AT^{2}+QAT)\) is more efficient than RBUA with \(O(QAT^{2})\). We can make an intuitive comparison from the numerical simulation given in Section 7. ```
1:Input:\(\mathbf{a},T,B_{S},G_{t}^{s}(t\in T),\sigma_{t}^{s}(t\in T),A_{s}*,P_{max}\)
2:Global variable: G
3:Initialization:
4:From \(RBU\{A_{s},T,B_{S},G_{t}^{s}(t\in T),\sigma_{t}^{s}(t\in T),A_{s}*,P_{max}\) to obtain \(\mathbf{\psi},\mathbf{X},\mathbf{Y}\)
5:For\(m=1\) to \(T\)do
6: Retrieve the effective users set \(A_{s}*\) of \(\psi[A,1,m]\) and the corresponding optimum solution \(\widehat{p}_{1}^{s},\widehat{p}_{2}^{s},\ldots,\widehat{p}_{T}^{s}\)
7:Add \((A_{s}*,\widehat{p}_{1}^{s},\widehat{p}_{2}^{s},\ldots,\widehat{p}_{T}^{s})\) to G
8:End for
9:Function RBPUA(\(\mathbf{p^{s}}\))
10:Get \((A_{s}*,\widehat{p}_{1}^{s},\ldots,\widehat{p}_{T}^{s})\) which maximizing \(W_{RBPA}^{s}(A_{s}*,p^{s})=\sum_{l=1}^{|A_{s}|}\sigma_{l}^{s}(A_{s}*,min\left\{ \widehat{p}_{1}^{s},p^{s}\right\})+D^{s}\) from G
11:Return\(min\left\{\widehat{p}_{1}^{s},p^{s}\right\},\ldots,min\left\{\widehat{p}_{T}^{s},p^{ s}\right\}\)
12:End Function RBPUA
```

**Algorithm 4** Improved scheme for single-resource-block power and user allocation (RBPUA)

## 6 Multi-Subchannel Resource Optimization

Based on the single-resource-block optimization, the multi-resource-block joint marriage user and power allocation problem can be defined by the following formula:

\[\varphi_{2}=\psi[a,\,n,\,m+1]. \tag{40}\]2

\[\varphi_{2}=\psi[a,\,n,\,m+1]. \tag{40}\]3

\[\varphi_{2}=\psi[a,\,n,\,m+1]. \tag{40}\]4

\[\varphi_{2}=\psi[a,\,n,\,m+1]. \tag{40}\]5

For power budget \(P^{s}\), the best resource allocation result can be obtained quickly for each resource by invoking the RBPUA scheme. Therefore, the power budget directly impacts the optimization solution of power and marine user allocation. \(W_{MRB}\) is ultimately the process of optimizing these two variables (\(\widehat{p}^{s}\) and \(A_{s}*\)) with \(P^{s}\) together. Suppose we can replace several definitions in \(W_{MRB}\) with \(W_{s}(p^{s})\) and conditions in problem \(W_{RB}^{s}\). In that case, \(W_{MRB}\) will be transformed into the equivalent problem \(W^{\bullet}\) put forward at the beginning of this paper. ### Heuristic Optimization Algorithm Based on Gradient Descent (GRAD)

The improved gradient descent algorithm combined with the RBPUA can effectively solve the joint power and sea user allocation problems. Hence, this paper regards the heuristic algorithm as the performance standard of the multi-resource-block joint optimization scheme. Compared with the traditional optimization algorithm, this scheme offers the advantage of reducing the complexity and achieving the optimal WAR. Its principle lies in the optimization of two stages. Specifically, the first stage follows the projected gradient descent of the power vector \(\mathbf{P^{s}}=\left(P^{1},P^{2},\ldots,P^{S}\right)\) in the search space formed by constraints (problem \(W_{MRB}\): C2 and C3). The second stage calculates \(W_{s}(p^{s})\) as well as its first derivative value \(W_{s}^{\prime}(p^{s})\). Then, these two values return to the first stage and participate in the judgment of the iteration conditions. After this, we select an appropriate step size to decide which point to start the next iteration. The step length \(\lambda\) is determined by the optimization of \(W_{s}\) along the ray \(\{\mathbf{P^{s}}+\lambda\bigtriangleup|\lambda\geq 0\}\). We adopt the precise straight-line search method in this paper. \[\varphi_{2}=\psi[a,\,n,\,m+1]. \tag{40}\]6Let \(P_{1}^{S}\), \(P_{2}^{S},\ldots,P_{T}^{S}\) represent the output power allocation value of the RBPUA algorithm.