Consider the problem below:

$mi{n}_{in\{0,1\}}ni=1$ vixi

s$.$t$.ni=1$ wixi $jn,1iv\frac{i}{w}i=v\frac{j}{w}j{f}^{**}{q}^{**}0f^{**}-q^{**}ma{x}_i=1$dotsvvi

where $wi,vi>0.Weaso$ know $jn,1$i dont exist such that $v\frac{i}{w}i=v\frac{j}{w}j.$

$a.$ Find the dual $of$ this problem and its solution.

$b.$ Let $f^{**}$ and $q^{**}be$ the $be$ the optimal solutions $of$ the primal and dual problem respectively.

Now Prove:

$0f^{**}-q^{**}ma{x}_i=1$dotsvi