Problem 20

Fix the Puiseux series field \(R = \mathbb{R}\{\!\{ \epsilon \}\!\}\). Pick a matrix \(A \in \mathbb{R}^{4 \times 8}\), column vector \(b \in R^4\) and row vector \(c \in R^8\), with entries that look random, with powers of \(\epsilon\). Solve the linear programming problem

Maximize \(c \cdot x\) subject to \(A x = b\) and \(x \geq 0\)

Solution

In Polymake:

application "polytope";
$x = new PuiseuxFraction<Min>(new UniPolynomial<Rational,Rational>('x'));
$A = new Matrix<PuiseuxFraction<Min>>([[2,3,$x*$x,11]]);
$c = new Vector<PuiseuxFraction<Min>>([0,1,2+$x]);
$P = new Polytope<PuiseuxFraction<Min>>(INEQUALITIES=>zero_vector|unit_matrix($A->cols-1),EQUATIONS=>$A);
print $P->LP(LINEAR_OBJECTIVE=>$c)->MAXIMAL_VALUE;