# Continuity of a Parametric Linear Program

Consider the convex optimization problem

$$\min_{x \in X, \ y \in Y } x$$

$$\text{sub. to } \ x A + B y + C = 0$$

where $X = [0,1] \subset \mathbb{R}$, $Y \subset \mathbb{R}^M$ are compact and convex sets and $p = (A,B,C)$ is a set of given parameters.

Let $x^*(p)$ the solution associated to parameters $p$.

I wonder if the solution is continuous with respect to the set of parameters:

$$\forall \epsilon > 0 \ \exists \delta>0 \text{ such that: } \ ||p - \tilde{p} || < \delta \ \Rightarrow \ || x^*(p) - x^*(\tilde{p}) || < \epsilon$$

In other words, is the solution of a Linear Programming (LP) problem continuous with respect to the parameters of the problem?

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Since it's supposed to be a convex problem, do you want $Y$ to be convex? –  Robert Israel Mar 28 '12 at 1:26

Hint: take $M=1$, $Y=[0,1]$, $B=1$, $C=-A$. What happens for $A>0$ and for $A <0$?