the following definition for developing the main results of this section.

*Definition 7.1*. Let $\phi:\IR^{n } \rightarrow \IR$ be a continuously differentiable convex function given by () , where $\theta_\rom{max} \in \IR$ is a projection norm bound imposed on $\theta \in \IR^n$ and $\varepsilon_\theta >0$ is a projection tolerance bound. Then, the *projection operator* $\rom{Proj}:\IR^{n } \times \IR^{n } \rightarrow \IR^{n }$ is defined by $$\begin{aligned}
   \Proj(\theta,y)
   \teq
   \left\{ \begin{array}{cl}
      \hspace{-2.8cm} y,  \ \ \ \mbox{ if $\phi(\theta) < 0$},
      \\
      \hspace{ 0.0cm} y, \ \ \ \mbox{ if $\phi(\theta) \ge 0$ and $\phi'(\theta)y  \le 0$},
      \\
      \hspace{-2.4cm} y - \frac{\phi'^\mathrm{^\rom{T}}(\theta)\phi'(\theta) y} {\phi'(\theta)\phi'^\mathrm{^\rom{T}}(\theta)} \phi(\theta), \\ \hspace{0.85cm} \mbox{ if $\phi(\theta) \ge 0$ and $\phi'(\theta)y>0 $},
   \end{array} \right.
   \label{my_proj_operator}\end{aligned}$$ where $y \in \IR^{n }$.

*Remark 7.1*. It follows from Definition 7.1 that (-\^\*)((,y)-y)0, \^\*\^n, \[defnt:b\] holds [@project]. The definition of the projection operator can be generalized to matrices as \_(,Y)=( (\_1 (), \_1 (Y)),  …,   (\_m (), \_m (Y)) ), where $\Theta \in \IR^{n \times m}$, $Y \in \IR^{n \times m}$, and $\rom{col}_i (\cdot)$ denotes $i$-th column operator. In this case, for a given $\Theta^* \in \IR^{n \times m}$, it follows from (\[defnt:b\]) that    =\_[i=1]{}\^m 0,\
holds. Throughout this section, we assume 