## 3 A Double Optimal Solution
### Mathematical Preliminaries  
To determine the unknowns \(\alpha_{j},\ j=0,1,\ldots,m\), the following main theorem is an extension of [39] to a non-square system (2). Theorem 3.1: _Assume that \(m_{0}=\text{rank}(\mathbf{A})\leq\min(n,q)\). \(\mathbf{U}\) satisfies Equation (25) with \(m<m_{0}\), \(\mathbf{x}\in\mathcal{K}_{m}^{\prime}\), approximately satisfying Equation (2) derived from two minimizations (40) and (41), is a double optimal solution (DOS), given by_  
\[\mathbf{x}=\alpha_{0}(\mathbf{u}_{0}-\mathbf{V}\mathbf{A}\mathbf{u}_{0}+ \lambda_{0}\mathbf{V}\mathbf{b}), \tag{42}\]  
_where_  
\[\mathbf{u}_{0}=\mathbf{A}^{\mathsf{T}}\mathbf{b},\ \mathbf{J}= \mathbf{A}\mathbf{U},\ \mathbf{C}=\mathbf{J}^{\mathsf{T}}\mathbf{J},\ \mathbf{D}=\mathbf{C}^{-1},\ \mathbf{V}=\mathbf{U}\mathbf{D}\mathbf{J}^{\mathsf{T}},\ \mathbf{P}=\mathbf{A}\mathbf{V}=\mathbf{J}\mathbf{D}\mathbf{J}^{\mathsf{T}},\] \[\lambda_{0}=\frac{\|\mathbf{A}\mathbf{u}_{0}\|^{2}-\mathbf{u}_{0}^ {\mathsf{T}}\mathbf{A}^{\mathsf{T}}\mathbf{P}\mathbf{A}\mathbf{u}_{0}}{ \mathbf{b}^{\mathsf{T}}\mathbf{A}\mathbf{u}_{0}-\mathbf{b}^{\mathsf{T}} \mathbf{P}\mathbf{A}\mathbf{u}_{0}}, \tag{43}\] \[\mathbf{w}=\lambda_{0}\mathbf{P}\mathbf{b}+\mathbf{A}\mathbf{u}_ {0}-\mathbf{P}\mathbf{A}\mathbf{u}_{0},\] (44) \[\alpha_{0}=\frac{\mathbf{w}^{\mathsf{T}}\mathbf{b}}{\|\mathbf{w} \|^{2}}. \tag{45}\]_To guarantee the existence of \(\mathbf{D}=\mathbf{C}^{-1}\), the dimension \(m\) of \(\mathcal{K}_{m}\) follows \(m<m_{0}\)._  
Proof. : In view of Equation (38), we can write  
\[\mathbf{b}^{\mathrm{T}}\mathbf{y}=\mathbf{b}^{\mathrm{T}}\mathbf{y}_{0}+ \mathbf{b}^{\mathrm{T}}\mathbf{J}\mathbf{\alpha}, \tag{46}\] \[\|\mathbf{y}\|^{2}=\|\mathbf{y}_{0}\|^{2}+2\mathbf{y}_{0}^{ \mathrm{T}}\mathbf{J}\mathbf{\alpha}+\mathbf{\alpha}^{\mathrm{T}}\mathbf{C} \mathbf{\alpha} \tag{47}\]  
for \(f\) in Equation (40), where  
\[\mathbf{C}:=\mathbf{J}^{\mathrm{T}}\mathbf{J} \tag{48}\]  
is an \(m\times m\) positive definite matrix; hence, \(\mathbf{C}^{-1}\) exists. For the rank deficiency matrix \(\mathbf{A}\), we suppose that its rank is \(m_{0}=\mathrm{rank}(\mathbf{A})\), where \(m_{0}<\min(n,q)\); to guarantee that \(\mathbf{C}\) is an \(m\times m\) positive definite matrix, \(m\) must be smaller than \(m_{0}\). The minimality condition of Equation (40) is  
\[(\mathbf{b}^{\mathrm{T}}\mathbf{y})^{2}\
abla_{\mathbf{\alpha}}\|\mathbf{y} \|^{2}-2\mathbf{b}^{\mathrm{T}}\mathbf{y}\|\mathbf{y}\|^{2}\
abla_{\mathbf{ \alpha}}(\mathbf{b}^{\mathrm{T}}\mathbf{y})=\mathbf{0}; \tag{49}\]  
\(\
abla_{\mathbf{\alpha}}\) is the gradient with respect to \(\mathbf{\alpha}\). Cancelling \(\mathbf{b}^{\mathrm{T}}\mathbf{y}\) in Equation (49) yields  
\[\mathbf{b}^{\mathrm{T}}\mathbf{y}\mathbf{y}_{2}-2\|\mathbf{y}\|^{2}\mathbf{y} _{1}=\mathbf{0}, \tag{50}\]  
where  
\[\mathbf{u}_{0}=\mathbf{A}^{\mathsf{T}}\mathbf{b},\ \mathbf{J}= \mathbf{A}\mathbf{U},\ \mathbf{C}=\mathbf{J}^{\mathsf{T}}\mathbf{J},\ \mathbf{D}=\mathbf{C}^{-1},\ \mathbf{V}=\mathbf{U}\mathbf{D}\mathbf{J}^{\mathsf{T}},\ \mathbf{P}=\mathbf{A}\mathbf{V}=\mathbf{J}\mathbf{D}\mathbf{J}^{\mathsf{T}},\]0 \[\mathbf{u}_{0}=\mathbf{A}^{\mathsf{T}}\mathbf{b},\ \mathbf{J}= \mathbf{A}\mathbf{U},\ \mathbf{C}=\mathbf{J}^{\mathsf{T}}\mathbf{J},\ \mathbf{D}=\mathbf{C}^{-1},\ \mathbf{V}=\mathbf{U}\mathbf{D}\mathbf{J}^{\mathsf{T}},\ \mathbf{P}=\mathbf{A}\mathbf{V}=\mathbf{J}\mathbf{D}\mathbf{J}^{\mathsf{T}},\]1  
From Equation (50), \(\mathbf{y}_{2}\) is observed to be proportional to \(\mathbf{y}_{1}\) in the \(m\)-dimensional space \(\mathbb{R}^{m}\), which is supposed to be  
\[\mathbf{u}_{0}=\mathbf{A}^{\mathsf{T}}\mathbf{b},\ \mathbf{J}= \mathbf{A}\mathbf{U},\ \mathbf{C}=\mathbf{J}^{\mathsf{T}}\mathbf{J},\ \mathbf{D}=\mathbf{C}^{-1},\ \mathbf{V}=\mathbf{U}\mathbf{D}\mathbf{J}^{\mathsf{T}},\ \mathbf{P}=\mathbf{A}\mathbf{V}=\mathbf{J}\mathbf{D}\mathbf{J}^{\mathsf{T}},\]2  
where \(2\lambda\) is a proportional factor determined below.