LP: how to understand Duality and simplex I am learning about Linear Programming right now.. I learned that we can use simplex to solve linear program and I also learned that every linear problem has a dual problem because of duality.. I am so confused now.. Since we can use simplex algorithm to solve Primal LP(get max or min), why should we transform Primal LP to Dual LP? I mean for what purpose, we should transform Primal LP to dual LP?
 A: PRIMAL: $$ \mbox{MAX } c x; \; x \geq 0, \;  A x \leq b.    $$
DUAL $$ \mbox{MIN } y b; \; y \geq 0, \; y A \geq c.    $$
With $c,y$ row vectors and all entries of $y$ non-negative, written $y \geq 0,$ and with $b,x$ row vectors and all entries of $x$ non-negative, written $x \geq 0,$ with $Ax \leq b$ and $yA \geq c,$ WEAK DUALITY is nothing more than this:
$$\color{magenta}{ c x \leq y A x \leq y b.}  $$ I wish people would tell me these things. 
So weak duality is nothing more than associativity of matrix multiplication and the fact that all entries of the column vector $x$ and the row vector $y$ are required to be non-negative. 
Strong duality is the fact that, if both problems are feasible, the optimal values of the objective functions ($cx, yb$) are the same. I will see if I can find a one-line proof of that. But it may be more difficult. 
In the end, I suspect the dual problem is for sensitivity analysis: what happens if some of the entries in $A$ or $b$ or $c$ are changed a little? This is a reasonable question; all of this is done on computer, and everything is a decimal (well, binary) approximation. 
