Question

Hi all. Suppose I have a linear programming problem on the vector variable $x$ that has many solutions and let $U$ be the set of these solutions. Suppose I have a second LP problem on $y \in U$. Therefore, I have a cascaded optmization problem. Is there anyway to combine them in a single optimization problem?

Answer 1

You can solve the first LP, obtain the optimal value, and then add a constraint that the first objective has that optimal value. Then you can add your second set of constraints to the original set of constraints and solve that LP. e.g. if your original LP is

$\max c^{T}x $

subject to

$Ax=b$

$x \geq 0$

Suppose the optimal objective value is $p^{*}$. Now suppose you want to solve a second problem

$\max d^{T}x $

subject to

$ Fx=g$

$ x \in U$

This can be formulated as

$\max d^{T}x $

subject to

$Ax=b$

$c^{T}x=p^{*}$

$Fx=g$

$x \ge 0$

Most simplex based LP codes can efficiently reoptimize after adding the additional constraints.