Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

$A \lambda A^{T} $ (quadratic form?) is used with matrices to check definiteness. What about with vectors? If I see conditions such as $\bar{x} > 0$, how can I know whether it means $\bar{x}_{i} > 0 \quad \forall i \in I$ or $\bar{x}:=\bar{x} \lambda \bar{x}^{T}>0$?

I am trying to understand the conditions in the standard form integer optimization problem where I see such cases:

$$ \text{minimize }\ \ \bar{c}^{T}\bar{x} + \bar{d}^{T}\bar{y}$$

so that $$A\bar{x} + B\bar{y} = b$$ $$\bar{x}, \bar{y} \geq 0$$ and $\bar{y}$ is continuous.

share|improve this question
    
I've honestly never seen the notation $\vec{x}>0$ before. What kind of problem are you seeing it in? –  anon Sep 22 '11 at 10:15
1  
I'm guessing they mean $x_i>0$ for each $i$, as that would be a standard optimization constraint while the quadratic constraint seems unnatural. But like I said, I'm guessing. –  anon Sep 22 '11 at 10:29
    
@anon: I am trying to translate the lecture slide 3 into English here. Originally, they apparently mean that $x \in \mathbb Z$ but I see no point otherwise to transpose the other vars such as $\bar{c}$. So I think they mean $\bar{x} \in \mathbb Z^{n}$. Anyway this is apparently copycatted from Introduction to Linear Optimization -book, have to get it to understand the definition here. I think it should be some standard def for ILO problem. –  hhh Sep 22 '11 at 10:30

1 Answer 1

up vote 1 down vote accepted

In the context of linear (or mixed integer) programming the inequality $x\geq 0$ is always meant in the "componentwise" sense, i.e. $x_i\geq 0$ for all $i$.

Note that for matrices one has to be even more cautious as there are the notions of "positive definite" (of denoted by $A\succ 0$), "positive" or even "total positive", which are all different.

In the context of semidefinite programming one often has both positivity constraints for vectors (something denoted by $Ax\geq 0$) and constraints on positive semidefiniteness (something denoted by $B\succcurlyeq 0$).

share|improve this answer

Your Answer

 
discard

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.