My question is about the geometrical solution for LP: how do we know that the level set function is decreasing or increasing?
Where did we get the -1/2 from?
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The $- \frac12$ is just an example value, for which the level set is plotted. You could have taken any other value and have plotted an according level set (as the question asks you).
We know that the steepest gradient is orthogonal to the level sets. The question is which direction is the decreasing one. You can see that you have to move the plotted level set in the direction of the given arrow, as this corresponds to decreasing $x_1$ and increasing $x_2$, i.e. decreasing the function $x_1 - x_2.$ If you move in the opposite direction, you increase the function.
blat
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