Conditional constraint activated by binary variable

Question

For each time step $t$, $T_1(t),...,T_n(t)$ are continuous variables, $z(t)$ are binary variables. $T_c(t)$ is known. I am trying to express the following constraint in a Linear Programm. $$ T_i(t+1) = \left\{\begin{array}{ll}T_c(t) & \text{if } z_t = 1\\T_i(t) & \text{if } z_t = 0\end{array}\right. $$

Any hints? Many Thanks

Answer 1

You have the bilinear equality constraint $T_i(t+1) = z_t T_c(t) +(1-z_t)T_i(t)$. In this, you can linearize the binary times continuous expression using a standard big-M model.

https://or.stackexchange.com/questions/39/how-to-linearize-the-product-of-a-binary-and-a-non-negative-continuous-variable