凸优化中常用算法的计算复杂度

• 线性规划（LP）
• 逐次凸逼近（SCA）
• 块坐标下降（BCD）
• 二分法 （Bisection）
• 穷举法
• 参考文献

线性规划（LP）

As explained in [12], the complexity of a standard linear problem is of order O ( a 2 b ) \mathcal{O} (a^2b) O(a2b), where a a a is the number of variables and b b b is the number of constraints.
For instance, the problem (P2.1) in ref [2], it is easy to show that the complexity of (P2.1) is of order O ( ( M N + K N + K + K M N ) ( K M ) 2 ) \mathcal{O} \left ( (MN + KN + K+ KMN)(KM)^2 \right) O((MN+KN+K+KMN)(KM)2).
Note that (P2.1) is a linear function which only consists of the scheduling variables.

• 变量个数：KM，为什么不是KMN？
• 约束(10) 的个数 M N + K N MN+KN MN+KN
• 约束(11) 的个数 K K K
• 约束(14) 的个数 K M N KMN KMN

逐次凸逼近（SCA）

SCA算法的计算复杂度和 “迭代次数（ L L L）”，以及每次迭代中的 “更新变量的个数（ N K NK NK）”有关；计算复杂度的表达式为 O ( N K L ) \mathcal{O} (NKL) O(NKL).

块坐标下降（BCD）

BCD algorithm 即 Alternating algorithm.
Alternating/BCD算法的收敛 要求每一个块的优化要达到最优解，而实际问题中每个块问题往往是非凸的，用凸优化技术SCA求解得到的往往是次优解，所以在证明算法的收敛性的时候，需要自行证明一下优化目标函数的非减、非增，证明是存在 upper bound 或 lower bound，如此可证明问题收敛。
在保证问题收敛后，块问题中有多个优化变量时的计算复杂度和SCA算法一样；只有一个优化变量时，变量数为 N N N，则计算复杂度为 O ( N 3.5 ) \mathcal{O} (N^{3.5}) O(N3.5).

二分法 （Bisection）

二分法的计算复杂度是 O ( log ⁡ 2 ( L 0 / e ) ) \mathcal{O} (\log_2(L_0/e)) O(log2​(L0​/e)) , 其中 L 0 L_0 L0​ 是最初的长度间隔， e e e 是要求的精度.

穷举法

穷举法的计算复杂度就是穷举次数。

参考文献

[1] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004.
[2] A. Bejaoui, K. Park and M. Alouini, “A QoS-Oriented Trajectory Optimization in Swarming Unmanned-Aerial-Vehicles Communications,” in IEEE Wireless Communications Letters, vol. 9, no. 6, pp. 791-794, June 2020, doi: 10.1109/LWC.2020.2970052.
[3] W. Luo, Y. Shen, B. Yang, S. Wang and X. Guan, “Joint 3-D Trajectory and Resource Optimization in Multi-UAV-Enabled IoT Networks With Wireless Power Transfer,” in IEEE Internet of Things Journal, vol. 8, no. 10, pp. 7833-7848, 15 May15, 2021, doi: 10.1109/JIOT.2020.3041303.
[4] C. Zhan and Y. Zeng, “Aerial–Ground Cost Tradeoff for Multi-UAV-Enabled Data Collection in Wireless Sensor Networks,” in IEEE Transactions on Communications, vol. 68, no. 3, pp. 1937-1950, March 2020, doi: 10.1109/TCOMM.2019.2962479.
[5] R. Duan, J. Wang, C. Jiang, H. Yao, Y. Ren and Y. Qian, “Resource Allocation for Multi-UAV Aided IoT NOMA Uplink Transmission Systems,” in IEEE Internet of Things Journal, vol. 6, no. 4, pp. 7025-7037, Aug. 2019, doi: 10.1109/JIOT.2019.2913473.
[6] Q. Hu, Y. Cai, A. Liu, G. Yu and G. Y. Li, “Low-Complexity Joint Resource Allocation and Trajectory Design for UAV-Aided Relay Networks With the Segmented Ray-Tracing Channel Model,” in IEEE Transactions on Wireless Communications, vol. 19, no. 9, pp. 6179-6195, Sept. 2020, doi: 10.1109/TWC.2020.3000864.