Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 1 / 7H (Optimisation)

Multiplying $Ax\le b$ by $y\ge 0$ gives the upper bound $c^{T}x\le y^{T}Ax\le y^{T}b$ whenever $A^{T}y\ge c$. Minimising the bound yields the dual

The primal point $x=(1/2,1/2,0)$ is feasible and has value $5/2$. The dual point $y=(1/2,3/2)$ is feasible becauseand has value $2(1/2)+3/2=5/2$. Weak duality proves both are optimal. Thus the requested optimal primal solution is