Difference of Convex Relaxation (DC)

问题背景

$$\begin{array}{rl}& \underset{m}{\mathrm{minimize}}\Vert \mathbf{m}{\Vert }^2\\ & \mathrm{subject}\mathrm{to}\Vert {\mathbf{m}}^{\mathsf{H}}{\mathbf{h}}_k^e{\Vert }^2\ge 1,\forall k.\end{array}$$

$$\begin{array}{rl}{\mathcal{P}}_1:& \underset{M}{\mathrm{minimize}}\mathrm{Tr}(M)\\ & \mathrm{subject}\mathrm{to}\mathrm{Tr}(\mathbf{M}{\mathbf{H}}_k)\ge 1,\forall k,\\ & \mathbf{M}⪰0,\mathrm{rank}(\mathbf{M})=1,\end{array}$$

$$\begin{array}{rl}\text{find}& v\\ \text{subject to}& {\tilde{v}}^{\mathsf{H}}{\mathbf{R}}_k\tilde{\mathbf{v}}+|c_k{|}^2\ge 1,\forall k,\\ & |v_n{|}^2=1,\forall v=1,\cdots ,N,\end{array}$$

$$\begin{array}{rl}{\mathcal{P}}_2:\mathrm{find}\mathbf{V}& \\ \mathrm{subject}\mathrm{to}& \mathrm{Tr}({\mathbf{R}}_k\mathbf{V})+|c_k{|}^2\ge 1,\forall k,\\ & {\mathbf{V}}_{n,n}=1,\forall n=1,\cdots ,N,\\ & \mathbf{V}⪰0,\mathrm{rank}(\mathbf{V})=1.\end{array}$$

为了进一步解决问题P1和问题P2中的非凸性，一种流行的方法是通过SDR技术简单地丢弃非凸的秩一约束[17]。由此产生的SDP问题可以通过现有的凸优化解算器（例如CVX [19]）来有效地求解。如果SDP问题的最优解是秩一的，则可以通过秩一分解来恢复原始问题的最优解。另一方面，如果SDP问题的最优解不是秩1，则需要应用高斯随机化[17]等附加步骤来提取原始问题的次优解。然而，观察到对于高维优化问题（例如，天线数量N增加），则对于SDR方法返回秩一解的概率变低，这产生显著的性能恶化[7]、[18]。为了克服SDR方法的局限性，我们在下面的章节中提出了一种新的DC框架来解决问题P1和问题P2。

所针对的一般问题

一阶约束问题的DC框架

为了便于介绍，我们首先考虑具有秩1约束的一般低秩矩阵优化问题的DC算法，如下所示，

$$\begin{array}{rl}& \underset{\mathbf{X}\in \mathcal{C}}{\mathrm{minimize}}\mathrm{Tr}({\mathbf{A}}_0\mathbf{X})\\ & \mathrm{subject}\mathrm{to}\mathrm{Tr}({\mathbf{A}}_k\mathbf{X})\ge d_k,\forall k,\\ & \mathbf{X}⪰0,\mathrm{rank}(\mathbf{X})=1,\end{array}\text{\hspace{1em}(18)}$$

其中约束集$\mathcal{C}$是凸的。关于rank-one约束的一个关键观察是，它可以等效地写为DC函数约束，这在以下Proposition中正式陈述。

Proposition 1. For positive semidefinite (PSD) matrix $X\in {\mathbb{C}}^{N\times N}$ and $\mathrm{Tr}(\mathbf{X})\ge 1,\text{we have}$
$$\mathrm{rank}(\mathbf{X})=1⟺\mathrm{Tr}(\mathbf{X})-\Vert \mathbf{X}{\Vert }_2=0,\text{\hspace{1em}(19)}$$

$\begin{array}{rl}& where\text{ trace norm }\mathrm{Tr}(\mathbf{X})=\sum _{i=1}^N{\sigma }_i(\mathbf{X})\text{ and spectral norm}\\ & \Vert \mathbf{X}{\Vert }_2={\sigma }_1(\mathbf{X})\text{ with }{\sigma }_i(\mathbf{X})\text{ denoting the i-th largest singular}\\ & \text{value of matrix X.}\end{array}$

为了增强问题（18）的低秩解，我们提出将（19）中的DC函数作为惩罚分量添加到目标函数中，而不是经由SDR方法移除非凸秩一约束，从而得到

$$\begin{array}{rl}& \underset{X\in \mathcal{C}}{\mathrm{minimize}}\mathrm{Tr}({\mathbf{A}}_0\mathbf{X})+\rho \cdot (\mathrm{Tr}(\mathbf{X})-\Vert \mathbf{X}{\Vert }_2)\\ & \mathrm{subject}\mathrm{to}\mathrm{Tr}({\mathbf{A}}_k\mathbf{X})\ge d_k,\forall k,\\ & \mathbf{X}⪰0,\end{array}\text{\hspace{1em}(20)}$$

where $\rho >0$ is the penalty parameter. Note that we are able to obtain an exact rank-one solution $X^{\ast }$ when the nonnegative component (Tr$({\mathbf{X}}^{\ast })-\Vert {\mathbf{X}}^{\ast }{\Vert }_2)$ in the objective function is enforced to be zero.

DC算法

虽然问题（20）仍然是非凸的，但它可以通过利用优化最小化技术(MM方法，见我之前的博客)以迭代方式求解，从而产生DC算法[20]。主要思想是通过线性化凹项$-\rho \Vert X{\Vert }_2$将问题（20）转化为一系列简单的子问题X2、目标函数。具体地说，我们需要在第$t$次迭代时求解由下式给出的子问题：

$$\begin{array}{rl}& \underset{\mathbf{X}\in \mathcal{C}}{\mathrm{minimize}}\mathrm{Tr}({\mathbf{A}}_0\mathbf{X})+\rho \cdot ⟨\mathbf{X},\mathbf{I}-\partial \Vert {\mathbf{X}}^{t-1}{\Vert }_2⟩\\ & \mathrm{subject}\mathrm{to}\mathrm{Tr}({\mathbf{A}}_k\mathbf{X})\ge d_k,\forall k,\\ & \mathbf{X}⪰0,\end{array}\text{\hspace{1em}(21)}$$

where $X^{t-1}$ is the optimal solution of the subproblem at iteration $t-1.$ It is clear that the subproblem (21) is convex and can be solved efficiently by existing solvers such as CVX [19]. In addition, the subgradient $\partial \Vert \tilde{X}{\Vert }_2$ can be computed efficiently by the following proposition [3].

Proposition $2.\text{ For given PSD matrix }\mathbf{X}\in {\mathbb{C}}^{N\times N},thesub$- gradient $\partial \Vert \mathbf{X}{\Vert }_2\text{can be computed as }{v}_1{\mathbf{v}}_1^{\mathrm{H}},where{\mathbf{v}}_1\in {\mathbb{C}}^N$ is the leading eigenvector of matrix X.

所提出的DC算法从任意初始点收敛到问题（20）的临界点[20]。因此，我们在算法1中总结了所提出的DC算法。

Proposed Alternating DC Approach

In this subsection, we apply the proposed DC framework to problem ${\mathcal{P}}_1$ and problem ${\mathcal{P}}_2.$ Specifically, to find a rank-one solution to problem ${\mathcal{P}}_1$, we propose to solve the following DC programming problem

$$\begin{array}{rl}& \underset{M}{\mathrm{minimize}}\mathrm{Tr}(\mathbf{M})+\rho (\mathrm{Tr}(\mathbf{M})-\Vert \mathbf{M}{\Vert }_2)\\ & \mathrm{subject}\mathrm{to}\mathrm{Tr}({\mathbf{MH}}_k)\ge 1,\forall k,\\ & \mathbf{M}⪰0,\end{array}\text{\hspace{1em}(22)}$$

where $\rho >0$ is the penalty parameter. When the penalty component is enforced to be zero, problem (22) shall induce a rank-one solution $M^{\star }$, we can thus recover the solution $m$ to problem (12) through Cholesky decomposition $M^{\star }=\mathbf{m}{m}^{\mathrm{H}}.$

To detect feasibility for problem ${\mathcal{P}}_2$, we propose to minimize the difference between trace norm and spectral norm as follows,

$$\begin{array}{rl}& \underset{V}{\mathrm{minimize}}\mathrm{Tr}(\mathbf{V})-\Vert \mathbf{V}{\Vert }_2\\ & \mathrm{subject}\mathrm{to}\mathrm{Tr}({\mathbf{R}}_k\mathbf{V})+|c_k{|}^2\ge 1,\forall k,\\ & {\mathbf{V}}_{n,n}=1,\forall n=1,\cdots ,N,\\ & \mathbf{V}⪰0.\end{array}\text{\hspace{1em}(23)}$$

When the objective value of problem (23) becomes zero, we shall find an exact rank-one optimal solution $V^{\star }$. By Cholesky decomposition $V^{\star }=\tilde{v}\tilde{v}{\tilde{v}}^{\mathrm{H}}$, we can obtain a feasible solution $\tilde{v}$ to problem (14). If the objective value fails to be zero, we claim that problem ${\mathcal{P}}_2$ (i.e., problem (14)) is infeasible.

In summary, the proposed alternating DC algorithm for
solving problem $\mathcal{P}$ can be presented in Algorithm 2.