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# Introduction

Consider the following problem:

\begin{aligned}
P:= \min_{x_1,x_2} & f(x_1,x_2) \\\text{s.t. } x_1x_2 &\leq 1,\\ -10x_1-x_2 &\leq -6,\\ x_1 &\in [0,1.5],\\ x_2 &\in [1,4].
\end{aligned}

The feasible region for this problem corresponds to the red area in Figure 1, denoted as $$\Omega$$.

Notice that although the range of the variable $$x_1$$ is originally $$[0,1.5]$$, the feasible region in the figure only admits values of $$x_1$$ between $$0.2$$ and $$1$$ (red dashed lines in Figure 1). Knowing this, it is then logical to change the bounds of $$x$$ in the original problem from $$x_1 \in [0,1.5]$$ to $$x_1 \in [0.2,1]$$ (i.e., tightening the bounds of $$x_1$$).

This change will make other calculations and techniques used by Octeract Engine more effective (e.g., relaxations and branching).

In general, for a problem $$\min_{x \in \Omega} f(x)$$ we would like to find the tightest bounds for each variable $$x_i$$, i.e., find the lower bound $$l_i^\star = \min_{x \in \Omega} x_i$$, and upper bound $$u_i^\star = \max_{x \in \Omega} x_i$$.

However, unlike our previous example, it is not easy to solve the problems $$\min_{x \in \Omega} x_i$$ and $$\max_{x \in \Omega} x_i$$, and a different approach is required. Feasibility based bounds tightening (FBBT) , Optimisation based bounds tightening (OBBT) and Constraint propagation (CP), are some of the algorithmic techniques implemented in Octeract Engine to approach this important task.