All published worksheets from http://sagenb.org
Image: ubuntu2004
ISE@Lehigh University
Motivation
In Linear Optimization(LP), people can solve a given low-dimensional problem by simply scratching the feasible region after which we can immediately figure out the optimal solution(if it exists). Such method only solves LPs with variables because human beings are 3-dimensional animals. We have little sense of higher dimensions. However, we know that induction sometimes helps: by starting from lower-dimensional cases, we could infer some properties that a higher-dimensional one should also satisfy thus improve our understanding of all these problems. Furthermore, if we feed more possible lower-dimension cases to our brain, we magically increase our ability to discern those unforseen properties.
This demo aims to take one step toward this direction: let's take a look at the polyhedron as well as the optimal solution of a corresponding LP in 4-dimension. That is, the feasible region, which is a polyhedron, changes (linearly) as time elapses. Consider the LP in the form of:
Example
The feasible region is a polyhedron. For instance, the following example: $ \begin{eqnarray} \displaystyle{\min_{(x,t)\geq 0} } & -x_1 -6x_2 -13x_3 \\ \begin{align} \text{s.t.} \\ \ \\ \ \\ \ \\ \end{align} & \begin{align} x_1 + t &\leq 200\\ x_2 + t &\leq 300 \\ x_1 +x_2 +x_3 + t &\leq 400 \\ x_2 +3x_3 + t &\leq 600 \end{align} \end{eqnarray} t$.