# Help with rewriting a compound inequality

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Summary:
Help with rewriting optimality conditions for integer-convex functions.
See attached screenshot.
Stumped on this, I'll take anything at this point (hints, solution, etc).

PhDeezNutz

## Answers and Replies

BvU
Homework Helper
Big problems are to be split into smaller problems. You have two inequalities. Write the first one down and manipulate ! Then idem number two.

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Andrea94
BvU
Homework Helper
And: what does integer-convex mean for e.g. $$g(k+1) + g(k-1) - 2g(k) \quad \textsf{?}$$

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Big problems are to be split into smaller problems. You have two inequalities. Write the first one down and manipulate ! Then idem number two.

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Great, thanks! Didn't even think about the fact that I could do each inequality separately.

And: what does integer-convex mean for e.g. $$g(k+1) + g(k-1) - 2g(k) \quad \textsf{?}$$

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What do you mean? The expression you've written is the same thing as $$\Delta g(k) - \Delta g(k-1)$$ but I'm not sure how that is relevant.

BvU
Homework Helper
What do you mean? The expression you've written is the same thing as $$\Delta g(k) - \Delta g(k-1)$$ but I'm not sure how that is relevant.
Right, but ##\Delta g(k) - \Delta g(k-1)## can be ##\ge 0## or ##\le 0## for an integer-convex function ... whereas ...

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Right, but ##\Delta g(k) - \Delta g(k-1)## can be ##\ge 0## or ##\le 0## for an integer-convex function ... whereas ...

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Still not sure what I'm supposed to see here

BvU
Homework Helper
The problem with inequalities is that you can only multiply (or divide) left and right with something positive. As it happens, for integer-convex functions the sign of ##g(k+1) + g(k-1) - 2g(k)## is ...

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The problem with inequalities is that you can only multiply (or divide) left and right with something positive. As it happens, for integer-convex functions the sign of ##g(k+1) + g(k-1) - 2g(k)## is ...

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Hm, the only thing I can think of is that if ##k## is optimal and nonzero, then ##g(k+1) + g(k-1) - 2g(k)## is always positive since for optimal ##k## we have ##\Delta g(k) > 0## and ##\Delta g(k-1) < 0##. Is this what you mean?

BvU
Homework Helper
Not sure where your 'optimal' comes from (it seems to live in your context, but not in the context of this thread ?).

But: yes, for a convex function the second derivative is always positive and so is this ##g(k+1) + g(k-1) - 2g(k)##.

I figured it might help in manipulating the inequalities ...

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Andrea94
Not sure where your 'optimal' comes from (it seems to live in your context, but not in the context of this thread ?).

But: yes, for a convex function the second derivative is always positive and so is this ##g(k+1) + g(k-1) - 2g(k)##.

I figured it might help in manipulating the inequalities ...

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Ohh I see, thanks a lot for the help!