# dual variable with constraints dealing different time limit

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.

• I am new to OR but not to stats or engineering . What is RD ? I got that LB / UB are lower/upper bounds Sep 12 '21 at 22:09
• @WestCoastProjects it represents "Ramp Down" but I assume it is a not common sense in OR, so don't worry Sep 13 '21 at 5:50

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.
• Try writing out all the primal constraints of a small example (say $T=3$), associate a dual variable with each primal constraint, and then see what the dual constraints look like. Hopefully that will help. I don't know what you mean about "representation". The way you wrote your dual constraints looks fine to me; I just think there are errors in the middle one. Sep 13 '21 at 15:51