# Improved Branch-and-Bound Antenna Selection Algorithm for Massive MIMO
## 2 Representative Methods in Antenna Selection
### Random Search Algorithm  
This algorithm is an optimization method based on randomized strategies [28]. By randomly sampling in the solution space, the performance index of sampling points is evaluated; we then find the approximate optimal solution. In the antenna selection problem, the random search algorithm selects a subset of receiving antennas at random, evaluates its impact on system performance (such as channel capacity), and finally selects the optimal antenna subset. The search process is as follows:  
1. Initialization: Set the maximum number of iterations \(N_{iter}\) and the current optimal channel capacity \(C_{opt}=0\).
2. Iterative process: 1. Randomly select antenna subset: Randomly select \(M_{r}\) antennas among \(N_{r}\) antenna receiving antennas to form antenna subset \(S\). 2. Construct sub-matrix: According to the selected antenna subset \(S\), extract the corresponding sub-matrix \(\mathbf{H}_{s}\) from the channel matrix \(\mathbf{H}\). 3. Calculate channel capacity: Calculate channel capacity \(C_{s}\) corresponding to the current antenna subset. 4. Update best solution: if \(C_{s}>C_{opt}\), update \(C_{opt}=C_{s}\) and \(S_{opt}=S\).
3. Termination conditions: When the maximum number of iterations \(N_{iter}\) is reached, the search process is terminated; then, the optimal antenna subset \(S_{opt}\) and corresponding channel capacity \(C_{opt}\) are the output.  
The random search algorithm exhibits straightforward implementation that circumvents the need for intricate mathematical derivations, demonstrating applicability to antenna selection problems across diverse scales while maintaining inherent parallelization compatibility. However, due to its stochastic sampling mechanism, the algorithm may require extensive iterations to attain near-optimal solutions. The algorithmic performance proves critically dependent on the sampling quality, potentially resulting in solution quality instability. Furthermore, when applied to large-scale antenna configurations, the method may incur substantial computational resource demands due to its exhaustive search characteristics.  
Existing antenna selection algorithms face a trade-off between performance and computational complexity. While an exhaustive search guarantees global optimality, its combinatorial complexity grows exponentially with the number of antennas, rendering it impractical for a large-scale MIMO system [26]. Greedy algorithms reduce the computational load by iteratively selecting locally optimal antennas, yet they are prone to sub-optimal solutions with approximately 15% capacity loss [27]. The branch-and-bound (BAB) method approximates theoretical upper bounds through pruning strategies; the algorithm complexity is relatively low, but the implementation is complicated. Random search algorithms require substantial iterations to approach optimal solutions, while exhibiting unstable solution quality. Recent advancements in artificial intelligence (AI) have introduced novel approaches to antenna selection, such as deep learning-based classification methods [29; 30; 31; 32; 33; 34]. However, these techniques remain constrained by their dependence on extensive labeled datasets and limited model generalizability.