I would like to solve the following Regularized Least Squares Problem (Very Similar to LASSO): $$ \arg \min_{x} \frac{1}{2} {\left\| A x - b \right\|}^{2} + \lambda {\left\| x \right\|}_{1} $$ Where \( A \in {\mathbb{R}}^{m \times n} \) and \( b \in {\mathbb{R}}^{m} \). For \( x \in {\mathbb{R}}^{n} \) the solution can be achieved using Sub Gradient Method or Proximal Gradient Method. My question is, how can it be solved for \( x \in {\mathbb{C}}^{n} \) (Assuming \( A \in {\mathbb{C}}^{m \times n} \) and \( b \in {\mathbb{C}}^{m} \))? For instance, what is the Sub Gradient? Thank You. ## My Attempt for Solution 001The Gradient of \( f \left( x \right) \) is given by: $$ {\nabla}_{x} f \left( x \right) = {A}^{H} \left( A x - b \right) $$ The Sub Gradient of \( g \left( x \right) \) is given by: $$ {\partial}_{x} g \left( x \right) = \lambda \operatorname{sgn} \left( x \right) = \lambda \begin{cases} \frac{x}{ \left| x \right| } & \text{ if } x \neq 0 \ 0 & \text{ if } x = 0 \end{cases} $$ Namely it is the Complex Sign Function. Then, the Sub Gradient Method is given by: $$ {x}^{k + 1} = {x}^{k} - {\alpha}_{k} \left( {A}^{H} \left( A {x}^{k} - b \right) + \lambda \operatorname{sgn} \left( {x}^{k} \right) \right) $$ Where \( {\alpha}_{k} \) is the step size. Yet it won't converge to CVX Solution for this problem.
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Royi |