I have been reading the research paper ["Distributed Joint Power, Association and Flight Control for Massive-MIMO self-Organizing Flying Drones"][1](Paper is from IEEE so it might be stuck behind a pay wall) and I got stuck on the notation they use in there problem 2 (penal) statement. This problem seems to fall under electrical engineering and mathematics, so hopefully this fits here. The problem 2 (penal) is written as:
\begin{equation} \\Given:\hspace{90mm} x,y,z,\alpha, \mathbf{\widetilde{p}^{\mathit{(v)}}} \\\\\begin{matrix} Maximize: & \\ \mathbf{\widetilde{p}}_{\mathit{g}} \end{matrix} \mathit{U_g(\mathbf{\widetilde{p}}_{\mathit{g}};\mathbf{\widetilde{p}}_{-g}^{\mathit{(v)}})}+\sum_{g'\in \mathit{G}\backslash g} \bigtriangledown _{\mathbf{\widetilde{p}}_{g}} \mathit{U_{g'}(\mathbf{\widetilde{p}}^{\mathit{(v)}})(\mathbf{\widetilde{p}}_{g}-\mathbf{\widetilde{p}}_{g}^{\mathit{(v)}}) - \frac{\psi}{2}\left \| \mathbf{\widetilde{p}}_{g}-\mathbf{\widetilde{p}}_{g}^{\mathit{(v)}} \right \|}^{2} \\Subject \ to: \hspace{60mm} Constraints(24),(25) \end{equation}
p_gw which is part of the equation blow represents the transmit power that ground user g allocates to a specific pilot sequence. The reason a ground user can use multiple pilot sequences in this case is because the authors are converting an MINLP problem into a convex problem. To do this, they relax there pilot constraint and allow each user to allocate power to each pilot sequence.
\begin{equation} \mathbf{\widetilde{p}}_{g}^{\mathit{(v)}} = (log(\widetilde{p}_{gw}^{\mathit{(v)}}))^{w\in W}\hspace{60mm}(22) \\\mathbf{\widetilde{p}}_{-g}^{\mathit{(v)}} = (log(\widetilde{p}_{g'w'}^{\mathit{(v)}}))_{g'\in G \backslash g}^{w\in W} \hspace{60mm}(23) \end{equation}
The constraints are not important, I just need help understanding the notation in the middle equation. The authors called this equation the pricing-based solution algorithm where the main idea is to let each ground user iteratively determine its own pilot sequence.
The Utility equation is given below: \begin{equation} U_g(\mathbf{\widetilde{p}}_{g};\mathbf{\widetilde{p}}_{-g}^{\mathit{(v)}}) = \sum_{w\in \mathit{W}}C_{gw}(\mathbf{\widetilde{p}}_{g},\mathbf{\widetilde{p}}_{-g}^{\mathit{(v)}}) \end{equation} Where the capacity equation is given below: \begin{equation} C_{gw}(\mathbf{\widetilde{p}}) = Blog_2(1+\gamma_gw(\mathbf{\widetilde{p}}) ) \end{equation}
My thinking is that since each user can use multiple pilot sequences, p_g is a 1 x w vector where w is the total number of available orthogonal pilot sequences. I am also thinking that the utility equation in the center returns either a vector or a number. I mainly need help in understanding what the nabla does in the middle equation. I am relatively new to mathematical optimization, so I am not familiar with all the notation that is used. My first thought was that it is a gradient, but if p_g is a vector, does that mean the equation is taking the gradient of a vector? If anyone could help point me in the right direction, I would be very happy! I hope what I have written is clear and I can give more information if needed.
