Good evening. In lineary optimization, we have primal and corresponding dual programs. It is often said that the variables in the dual program can be interpreted as the shadow price for the corresponding constraint in the primal program. But it is never said WHY this is the case. Why do the variables in the dual program suddenly describe how the costs (or whatever you are trying to minimize) change when you increase the right hand side of a constraint by one? And furthermore, if the strong duality states, that the solution for the dual and primal are the same - how can be the optimal solution for the prime be the shadow price at the same time? They dont even necessarily have the same dimension?
Thanks in advance for every answer!