 Solution

For every primal feasible solution $\mathbf{x}$, weak duality givesThus no primal feasible solution improves on ${\mathbf{x}}^{\ast }$. Similarly, for every dual feasible solution $\mathbf{y}$,Therefore both candidates are optimal, with the same objective function value. Equality is a linear programming optimality certificate and does not require a separate existence argument.