Large-scale optimization problems are common in various fields such as engineering, finance, and computer science. These problems involve finding the optimal solution among a large number of candidate solutions, which can be computationally challenging. In this article, we propose two new algorithms that significantly reduce the computational complexity of solving these optimization problems.
First Algorithm: Active Reconfigurable Intelligent Surface (ARIS)
The ARIS algorithm is based on the concept of a reconfigurable intelligent surface. It uses a novel approach to solve optimization problems by iteratively updating the surface with new information. This process allows the algorithm to adapt to changing problem conditions and converge faster than traditional methods. We prove that the per-iteration complexity of ARIS is O(N 3 + M 2N + M 2K + N 2M + N 2K + M N K), which is much lower than existing algorithms.
Second Algorithm: Efficient Algorithm (EA)
The EA algorithm is a simplified version of the ARIS algorithm. It uses a closed-form update rule that reduces the computational complexity to O(N 3 + M 2N + M 2K). We show that EA is faster than existing algorithms when dealing with large problem sizes.
Simulation Results and Conclusion
We conduct extensive simulations to compare the performance of ARIS and EA with other algorithms. Our results show that both ARIS and EA are much faster than existing methods, especially for large problem sizes. Specifically, we find that ARIS is an order of magnitude faster than the next fastest algorithm when dealing with MIMO (Multiple Input-Multiple Output) and RIS (Reconfigurable Intelligent Surface) sizes beyond 100.
In conclusion, our proposed algorithms offer a significant improvement in computational efficiency for solving large-scale optimization problems. The ARIS algorithm provides the best performance among all the algorithms tested, while the EA algorithm offers a simpler and more efficient solution. These results demonstrate the potential of using reconfigurable intelligent surfaces to solve complex optimization problems.
