Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / T / T2 / a / iii

Take the dual vector $y=(8/17,45/17{)}^T$. Its column constraints have values $5,12,53/17,233/17$, exceeding or equaling $c=(5,12,2,5{)}^T$; its entries are positive. Thus it is dual feasible. The first two constraints are equalities while the last two have strictly positive slack, suggesting $x_3=x_4=0$. Solve the two primal equalities to get $x_1=5/17$, $x_2=3/17$, both positive.

The resulting feasible objectives agree:Weak duality proves optimality. Strict dual slack in columns three and four and positive dual entries force the two primal equalities, proving primal uniqueness; positive first two primal coordinates force the two independent dual equalities, proving dual uniqueness.