# Linear programming such that feasible solution gives optimal solution to another

Let us say that we have a linear program $$P={ (min C^tx | Ax=b,x\geq0)}$$
Assume that $$(P)$$ has an optimal solution. Write a system of linear equations and inequalities $$(P_1)$$ such that any feasible solution to $$(P_1)$$ gives us an optimal solution to $$(P)$$.

What I thought: I just added the complementary slackness condition along with the dual constraints to the primal constraints and solve it, as it will satisfy:

1.Primal feasibility
2.Dual feasibility
3.Complementary slackness
Hence will get the optimal solution as a feasible solution to the Linear program
but Later I noticed that if I include complementary slackness the constraints are non-linear. So any suggestion?

• If you have primal and dual feasibility then complementary slackness is equivalent to the statement that primal and dual objective values are equal. – Michal Adamaszek Nov 14 at 11:27
• but for optimallity dont we need the complementery slackness. – onlymath Nov 14 at 11:36
• math.stackexchange.com/questions/3370732/… – Kuifje Nov 14 at 12:56