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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2024

Paper 1

Introduce nonnegative dual variables $y$ and $z$ for the two constraints. The <dual of a minimization linear program in inequality form> is
$$
\boxed{
\begin{aligned}
\text{maximise}\quad&ty+rz\\
\text{subject to}\quad&b_i y+c_i z\leq a_i
\quad(1\leq i\leq n),\\
&y,z\geq0.
\end{aligned}}
$$
Indeed, if $x$ and $(y,z)$ are feasible, then
$$
ty+rz
\leq y\sum_i b_ix_i+z\sum_i c_ix_i
=\sum_i(b_i y+c_i z)x_i
\leq\sum_i a_ix_i,
$$
which is <weak duality>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 1 / 7H / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 1 / 7H / a / Solution

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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 1 / 7H / a / Solution

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