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Hi there some classmates and I are trying to understand a review question for a class on convex optimization.

Consider the following programming instance: $\text {min} (x_3+x_4) s.t. -x_1-x_3+x_4=1,-x_2+x_3-x_4=1$, $\text{x}\ge1$.

Which of the following vectors provides a Farkas-type infeasibility certificate for the above problem:

A. $y=(1,1)$

B. $y=(-1,-1)$

C. $y=(2,2)$

D. All of the above

The answer is said to be "D", but would you please explain why it is the case? Thank you in advance and we wish you a wonderful day!

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  • $\begingroup$ Is there a typo in your question? A. and B. are exactly the same. Also, it would help to provide your definition of a "Farkas-type infeasibility certificate." $\endgroup$ – Math1000 Oct 24 '19 at 4:43
  • $\begingroup$ Thank you I just fixed it! I found a note on Farkas certificate of feasibility, it is not as bad as previously thought: people.seas.harvard.edu/~yaron/AM221-S16/lecture_notes/… $\endgroup$ – mathnoob777 Oct 24 '19 at 5:55
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Note that if you find a vector $(y_1, y_2)$, the linear combination of your equations is

$$y_1(-x_1-x_3+x_4)+ y_2(-x_2+x_3-x_4)=y_1+y_2 \implies $$

$$ -y_1 x_1 -y_2 x_2 +(-y_1 +y_2)x_3 + (y_1 -y_2) x_4=y_1+y_2$$

Now, if the associated coefficients are nonpositive, but the right side of the equality is positive

$$-y_1\leq 0, -y_2\leq 0, -y_1 +y_2\leq 0, y_1 -y_2\leq 0 ~ and ~ y_1+y_2>0$$

your system of equations is infeasible due to $x_1\geq 1$ and $ x_2\geq 1$ [there is no way of a sum of nonpositive numbers to be positive]. Thus $y=y_1=y_2>0$ is a specific case where

$$-y x_1 -y x_2 =2y$$

is infeasible for all $y>0$

**It is the same to say $Ax=b$ is infeasible iff $\exists y, ~ yA\leq 0 ~ and ~ yb>0$ **

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