Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
Since it's supposed to be a convex problem, do you want $Y$ to be convex? – Robert Israel Mar 28 '12 at 1:26
up vote 1 down vote accepted

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.