I was working my final year project, and it was basically a replication of a past paper. So when I was working on the optimization part. I was a bit confused about some concepts mentioned in that paper:$L(P^{D}_{i},\lambda^{e}_{i},\alpha,\beta,\gamma,\delta,\in)=T^{D}_{i}-Q^{D}_{i}EC^{D}_{i}-\alpha(P_{i}^{D}-P_{max})-\beta(\lambda^{e}_{i}-1)+\gamma(T^{D}_{i}-T_{min}^{D})+\\\delta(T_{i}^{C}-T_{min}^{C})+\in(P_{i}^{R}-P^{1}_{th})$
Where $L$ is the Lagrange functions, $P_{i}^{D} $ is the power, $\lambda_{i}^{e}$ is the power splitting ratio ,$\alpha,\beta,\gamma,\in,\delta$ are the lagrange multipliers respectively. So, as I understand if I want to solve the Lagrange functions, I have six simultaneous equations which are partial derivatives of Lagrange functions with respect to $P^{i}_{D},\lambda^{e}_{i},\alpha,\beta,\gamma,\delta,\in$.
But in this paper, the authors first solve $P_{i}^{D}$ and $\lambda_{i}^{e}$ by solving equations:$\frac{\partial L(P^{D}_{i},\lambda^{e}_{i},\alpha,\beta,\gamma,\delta,\in)}{\lambda_{i}^{e}}=0$ and $\frac{\partial L(P^{D}_{i},\lambda^{e}_{i},\alpha,\beta,\gamma,\delta,\in)}{P_{i}^{D}}=0$.
For this part i think i understand, but then the authors solve the Lagrange multipliers by using gradient methods:
Where $\{X\}^{+}=max\{0,X\}$.
I am new to optimization theory, can anyone explain this to me?
