Folks, I working with a Mixed Integer Non-Linear Program (MINLP), in which I have a non-linear function I am trying to bound, i.e. defining lower- and upper-bounds (under- and over-estimators). Below follows the program I am referring to.
P-MINLP $= { x \in \mathbb{B}^{n m} | f_i(x) \leqslant \tau^i \quad \forall i \in \mathbb{N}^{*}_{\leqslant n} }$
With the following constraints,
- $f_i(x) = \sum_{j = 1}^m \frac{x^i_j}{\kappa_j - g_j(x)}$, $\forall i \in \mathbb{N}^{*}_{\leqslant n}$;
- $g_j(x) = \sum_{i = 1}^n C^i x^i_j \leqslant \kappa_j - \epsilon$, $\forall j \in \mathbb{N}^{*}_{\leqslant m}$.
And the following constants,
- $\epsilon > 0$ is a small number (for example $10^{-5}$);
- $C^i, \tau^i \in \mathbb{R}^{+}$, $\forall i \in \mathbb{N}^{*}_{\leqslant n}$; And
- $\kappa_j \in \mathbb{R}^{+}$, $\forall j \in \mathbb{N}^{*}_{\leqslant m}$.
Two straightforward, linear, bounds we can derive are:
- $L_i(x) = \sum_{j = 1}^m \frac{x^i_j}{\kappa_j}$, $\forall i \in \mathbb{N}^{*}_{\leqslant n}$;
- $U_i(x) = \sum_{j = 1}^m \frac{x^i_j}{\epsilon}$, $\forall i \in \mathbb{N}^{*}_{\leqslant n}$.
Such that:
$L_i(x) \leqslant f_i(x) \leqslant U_i(x)$, $\forall i \in \mathbb{N}^{*}_{\leqslant n}$.
I am trying to find stronger, tighter, linear, or convex, bounds for the functions $f$s, or even for their factors, such as functions $g$s. Also, I am trying to understand if the function $f$ is convex given the proposed bounds.
Thanks for your attention, case any point is not clear, please, let me know.