Dual variables associated with same equation for different time instants

Question

I have three equations that are essentially the same equation defined for three time instants. The equations are basically calculating the state of energy of an energy storage facility.

\begin{align} e(t) &= e(0)-d(t)+\eta \cdot c(t);&&t=1\\ e(t) &= e(t-1)-d(t)+\eta\cdot c(t);&&1<t<T\\ e(t) &= 0.5\cdot E;&&t=T \end{align}

$\eta$ and $E$ are parameters; the rest are variables. The equations are defining the value for the same variable $e(t)$. My question is do I include a single variable to define the dual variables for all the equations or I need to define three separate variables with one for each equation. Therefore, if I were to write KKT conditions for the complete formulation, only one or three dual variables will show up.

Answer 1

You do not have $3$ constraints, you have $T$ constraints.

For example, if $T=5$, then we have

\begin{align}e(1)&=e(0)-d(1)+\eta\cdot c(0)\tag{1}\\e(2)&=e(1)-d(2)+\eta\cdot c(1)\tag{2}\\e(3)&=e(2)-d(3)+\eta\cdot c(2)\tag{3}\\e(4)&=e(3)-d(4)+\eta\cdot c(3)\tag{4}\\e(4)&=0.5\cdot E\tag{5}\end{align}

You might like to use the same symbol but index it with time $t$, for example introduce the dual variables $y_t$ for the $t$-th constraint or introduce a vector $y \in \mathbb{R}^{T}$.