*3.3. Joint Dwell Time and Bandwidth Optimization Problem Solution*

The optimization problem proposed in Equation (45) is non-convex, containing two parameters *T*d *<sup>m</sup>*,*q*,*<sup>k</sup>* and β*m*,*q*,*k*. We can use the exhaustive method to solve it, which is simple but too inefficient. The genetic algorithm uses selection, cross and mutation operators for searching, which has a great global search ability. However, the local search ability of this genetic algorithm is weak. In contrast, most of the classical nonlinear algorithms adopt the means of the gradient method, which has a strong local search ability, while also possessing a weak global search ability. As a result, we will solve the problem in (45) by NPGA [29], which combines the global search ability of the genetic algorithm and the local search ability of the classical nonlinear programming algorithms. The flowchart of NPGA is shown in Figure 1:

**Figure 1.** The nonlinear programming-based genetic algorithm (NPGA) flowchart.

By working out the problem (45) for *<sup>Q</sup>* · <sup>C</sup>*<sup>M</sup> <sup>N</sup>* times, we can get all the optimal solutions of the dwell time with respect to different target and radar combinations in the constraint of \$*<sup>N</sup> i*=1 *u q <sup>i</sup>*,*<sup>k</sup>* = *<sup>M</sup>*. Then we can use the exhaustive method to obtain the optimal results of the dwell time and radar allocation index in the constraints of \$ *Q q*=1 *u q <sup>i</sup>*,*<sup>k</sup>* ≤ 1. However, the exhaustive method is complex and inefficient. As a result, we propose a radar node selection algorithm with lower computation complexity.

Assuming *M* = 2, which means that each target is fixed to be tracked by two radars at each moment. We define *Rl* = {*a*, *b*}(*l* = 1, 2, ... , *L*) as the combinations of the two radars in the radar network, where *L* = C2 *<sup>N</sup>* <sup>=</sup> *<sup>N</sup>*! (*N*−2)!2! . When the target *<sup>q</sup>* is illuminated by the *Rl* index radars, suppose *Sl*,*k*,*q*,min = % *T*d *a*,*q*,*k*,min&(*l*) + % *T*d *b*,*q*,*k*,min&(*l*) denotes the minimum dwell time which is solved in (45) through NPGA, where % *T*d *a*,*q*,*k*,min&(*l*) and % *T*d *b*,*q*,*k*,min&(*l*) denotes the dwell times of radar *a* and radar *b*, respectively. The minimum dwell time matrix **S***k*,min which is composed of *Sl*,*k*,*q*,min, is shown in Table 1.


**Table 1.** Minimum dwell time matrix for the fixed radar combination (*M* = 2).

Similar to the term *u q i*,*k* , we define a set of binary variables *U<sup>q</sup> <sup>l</sup>*,*<sup>k</sup>* ∈ {0, 1} to represent the radar combination selection index.

$$L\_{l,k}^q = \begin{cases} 1, \text{ if the } q \text{th radar is translated by the } l \text{th radar combination at time index } k\\ 0, \text{ otherwise} \end{cases} \tag{46}$$

Then the optimization model of the radar combination allocation index can be described as:

$$\begin{aligned} \min & \sum\_{q=1}^{Q} \sum\_{l=1}^{L} \mathsf{LI}\_{l,k}^{q} S\_{l,pk,\min} \\ & \text{s.t.} \begin{cases} \sum\_{l=1}^{L} \mathsf{LI}\_{l,k}^{q} = 1 \\ \sum\_{l=1}^{L} \mathsf{LI}\_{l,k}^{q} R\_{l} \cap \left( \bigcup\_{l=1}^{L} \mathsf{LI}\_{l,k}^{m} R\_{l} \right) = \mathcal{Q}\_{\prime} \,\forall r \neq m, r, m = 1, 2, \dots, Q \end{cases} \end{aligned} \tag{47}$$

where the first constraints imply that each target is tracked by a fixed radar combination at time index *k*, while the second one suggests that a single radar tracks at most one target at *k*. The solution method of (47) can be shown in Algorithm 1.

**Algorithm 1.** Radar allocation method

**Step (1):** Working out the problem in (47) *<sup>Q</sup>* · *<sup>N</sup>*! (*N*−2)!2! times, then we can get the minimum dwell time matrix *q*

**<sup>S</sup>***k*,min in the constraint of \$*<sup>N</sup> i*=1 *u <sup>i</sup>*,*<sup>k</sup>* = 2.

**Step (2):** Sort the columns of matrix **S***k*,min in ascending order and assign the target corresponding to the smallest element in the first row to the corresponding radar combination.

**Step (3):** Remove the column vectors corresponding to the target assigned in Step (2). Remove all the row vectors of the radar which is contained in the radar combination assigned in Step (2).

**Step (4):** Repeat Step (2) and Step (3) until all the targets are assigned in order to obtain the optimal allocation matrix **U***k*,opt.

By using the above algorithm, we can obtain the optimal radar allocation results **U***k*,opt, where **U***k*,opt = 4 **U**1 *<sup>k</sup>*,opt, **<sup>U</sup>**<sup>2</sup> *<sup>k</sup>*,opt, ... , **<sup>U</sup>***<sup>Q</sup> k*,opt5 , **U***<sup>q</sup> <sup>k</sup>*,opt = 4 *Uq* 1,*k*,opt, *<sup>U</sup><sup>q</sup>* 2,*k*,opt, ... , *<sup>U</sup><sup>q</sup> L*,*k*,opt5<sup>T</sup> . When *U<sup>q</sup> <sup>l</sup>*,*k*,opt = 1, *<sup>u</sup> q <sup>a</sup>*,*k*,opt = *u q <sup>b</sup>*,*k*,opt <sup>=</sup> 1, *<sup>T</sup>*<sup>d</sup> *<sup>a</sup>*,*q*,*k*,opt = % *T*d *a*,*q*,*k*,min&(*l*) , *T*<sup>d</sup> *<sup>b</sup>*,*q*,*k*,opt = % *T*d *b*,*q*,*k*,min&(*l*) . When *U<sup>q</sup> <sup>l</sup>*,*k*,opt = 0, *<sup>u</sup> q <sup>a</sup>*,*k*,opt = *<sup>u</sup> q <sup>b</sup>*,*k*,opt = 0, *T*d *<sup>a</sup>*,*q*,*k*,opt <sup>=</sup> *<sup>T</sup>*<sup>d</sup> *<sup>b</sup>*,*q*,*k*,opt <sup>=</sup> 0. Then we can get **<sup>u</sup>***k*,opt and **<sup>T</sup>**<sup>d</sup> *<sup>k</sup>*,opt at time index *k*, which are the radar allocation index and dwell time optimization results, respectively.

The computational complexity of 0 is *O* % *Q*2 <sup>2</sup> <sup>×</sup> *<sup>N</sup>*! (*N*−2)!2! log2 % *<sup>N</sup>*! (*N*−2)!2!&&, while the computational complexity of the exhaustive method is *O* % *<sup>N</sup>*! (*N*−2)!2!&*Q* . Compared with the enumeration method, 0 can greatly reduce the computational complexity and improve the real-time performance.
