Is the Objective Function Always Nonnegative in Linear Optimization Problems?

[flyingpig]

Homework Statement

Given the follow LOP P

I am just going to write down the obj function because that is most important for my question and the constraints aren't

$$w = y_1 - 2y_2 + y_3$$

I was asked to show that $$y = (2t, 3t,t)^t$$ is a solution for all $$t\geq 0$$

So w = 2t - 2(3t) +t = 2t - 6t + t = -4t + t = -3t

Now initally I thought that as $$t \to \infty$$, $$-3t \to -\infty$$

I checked the key provided by my prof and he took $$t \to -\infty$$ and $$-3t \to \infty$$

Is it because we always assume w >0?

The flaw I made is that I never consider $$t \to \pm \infty$$.

Homework Equations

The Attempt at a Solution

 

[Mark44]

In applied problems, the objective function is usually nonnegative, but in more theoretic presentations, I don't see why this needs to be true.

 

[flyingpig]

In applied problems, the objective function is usually nonnegative, but in more theoretic presentations, I don't see why this needs to be true.

I asked my prof today and he kinda said the same thing about "yes intuitively that is right, we want obj f > 0". Then he added a bunch of things that confused me even more...

He stated something like this

max z = -min(-z)