Strong Duality Theorem in Linear Programming

[math8]

If a linear Program (P) has a feasible solution [itex]x_{o}[/itex], ( [itex]x_{o}[/itex] not necessarily optimal),does it follow that there exists a feasible solution to the dual problem (D) as well? If yes, why?

I know that the Strong Duality Theorem guarantees an optimal finite solution to the dual problem if the primal problem has an optimal finite solution. But I cannot see why this would be the case if the feasible solution to the primal is not necessarily optimal.

 

[Ray Vickson]

If a linear Program (P) has a feasible solution [itex]x_{o}[/itex], ( [itex]x_{o}[/itex] not necessarily optimal),does it follow that there exists a feasible solution to the dual problem (D) as well? If yes, why?

I know that the Strong Duality Theorem guarantees an optimal finite solution to the dual problem if the primal problem has an optimal finite solution. But I cannot see why this would be the case if the feasible solution to the primal is not necessarily optimal.

If P is the primal and D is the dual, the possibilities are: (i) P and D are both feasible (in which case they both have finite optimal solutions with equal objectives); (ii) P is feasible and D is infeasible (in which case P has no finite optimum); (iii) D is feasible and P is infeasible (in which case D has no finite optimum); (iv) both P and D are infeasible.

RGV

 

[math8]

Thanks :). That helps!