I ve two questions.
First . How to distinguish between nonlinear and linear
The difference between linear and nonlinear as I know is whether it is proportional to the result or not.
(that is linear is Linear equation )
second . Is it nonlinear to use an index as a variable in a MIP model?
for example)
$$sp_i \cdot y_{i,f} \le c_{f,t}\quad\forall i,j\quad\text{and}\quad t=S_i,\cdots,C_i.$$
$$S_i,C_i \ : variables , t: index .$$
If so, can you tell why?
thanks for reading.
A linear form in the variables $x_i$ is $\sum a_i x_i$ where $a_i$ are arbitrary constants. If you have products or powers of $x_i$ it is non-linear.
Some non-linear functions, like $|a_i x_i + b_i|$, for example, you can express in linear form using tricks.
Index in a MIP does not necessarily make it nonlinear. For example, the sum $$ \sum_{k=1}^5 kx_k = x_1 + 2x_2 + \ldots 5x_5 $$ is a valid linear combination of the variables, and so is the constraint $$ \sum_{k=1}^5 kx_k \le \sum_{i=3}^6 iy_i + \sum_{i=1}^{10}i^2, $$ since $\sum_{i=1}^{10}i^2$ is constant in the variables $x_i,y_i$ (but non-linear in $i$, which does not matter for the problem since we are only concerned with linearity in variables). But $\sum x_k^k$ would be nonlinear when $k \not \in \{0,1\}$.