dual variable with constraints dealing different time limit

Question

I am working on a unit commitment problem where I need to turn the constraints to dual. But the constraints do not deal with the same time period which makes dual process confusing. for a simple example:

$min \sum_{t=1}^T(Cx(t))$

$s.t.$

$ \quad x(t) \ge LB, -x(t) \ge -UB,\quad \forall t$

$ \quad x(t)-x(t-1) \ge -RD,\quad 1< t \le T$

$ \quad x(1) \ge -RD+x(0),\quad \quad t = 1$

$ \quad x(t) = D(t),\quad \forall t$

where $x(t)$ is the only variable here and $x(0)$ serves as a known initial condition. and below is my dual form of the problem and the $ \lambda_1(t),\ldots,\lambda_5(t)$ represent the dual multiplier of the above constraints. Note: for simplification, I ignore the units number here.

$max \sum_{t=1}^T(LB*\lambda_1(t))+\sum_{t=1}^T(-UB*\lambda_2(t)) + \sum_{t=2}^T(-RD*\lambda_3(t)) +(-RD+x(0))*\lambda_4+ \sum_{t=1}^T(D(t)*\lambda_5(t))$

$s.t.$

$ \quad\lambda_1(1) -\lambda_2(1) - \lambda_3(1) + \lambda_4 + \lambda_5(1) \le C$

$\quad \lambda_1(t) -\lambda_2(t) - \lambda_3(t) + \lambda_3(t-1) + \lambda_5(t) \le C, \quad 2 < t < T $

$\quad \lambda_1(T) -\lambda_2(T) + \lambda_3(T) + \lambda_5(T) \le c$

In most of the paper I read, authors either just ignore the initial condition or just use the second dual constraints above without consider the boundary of the constraints. The reason I write the constraints this way is that, for example, $T=4$ the third constraint can be found like the matrix below.

$\begin{bmatrix}-1&1&0&0\\0&-1&1&0\\0&0&-1&1\end{bmatrix}$

Is my formulation correct? or is there any more common mathematical representation of this type of problem? any advice will be appreciated! And sorry for poor English.

Answer 1

You're close, but your indexing is off in the middle dual constraint. I assume you mean $t\ge 2$ rather than $t>2$ in the second dual constraint (and that $c$ is $C$ in the last one). Keep in mind that $x(\tau)$ appears in the third primal constraint with coefficient +1 when $t=\tau$ and in with coefficient -1 when $t=\tau+1$, not when $t=\tau-1$ and $t=\tau$ respectively.