Help with rewriting a compound inequality

In summary, the conversation discusses splitting big problems into smaller ones and manipulating inequalities. The concept of integer-convex functions is also brought up, with the example of $$g(k+1) + g(k-1) - 2g(k)$$ being discussed. It is mentioned that for convex functions, the second derivative is always positive and this can help in manipulating inequalities.
  • #1
Andrea94
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TL;DR 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).

help.png
 
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  • #2
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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  • #3
And: what does integer-convex mean for e.g. $$g(k+1) + g(k-1) - 2g(k) \quad \textsf{?} $$

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  • #4
BvU said:
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.
 
  • #5
BvU said:
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.
 
  • #6
Andrea94 said:
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 ... :wink:

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  • #7
BvU said:
Right, but ##\Delta g(k) - \Delta g(k-1)## can be ##\ge 0## or ##\le 0## for an integer-convex function ... whereas ... :wink:

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Still not sure what I'm supposed to see here 😅
 
  • #8
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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  • #9
BvU said:
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 ...

##\ ##
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?
 
  • #10
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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  • #11
BvU said:
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 ...

##\ ##
Ohh I see, thanks a lot for the help!
 

1. What is a compound inequality?

A compound inequality is an inequality that contains two or more inequalities joined together by the words "and" or "or".

2. Why would I need to rewrite a compound inequality?

Rewriting a compound inequality can make it easier to solve or graph, especially if the original inequality is complex or difficult to understand.

3. How do I rewrite a compound inequality?

To rewrite a compound inequality, you must first identify the individual inequalities within the compound inequality. Then, you can use the properties of inequalities (such as addition, subtraction, multiplication, and division) to manipulate the inequalities until they are in a simpler form.

4. Can I rewrite a compound inequality without changing its meaning?

Yes, as long as you use the properties of inequalities correctly, you can rewrite a compound inequality without changing its meaning. This means that the solution set will remain the same.

5. Is there a specific order in which I should rewrite a compound inequality?

No, there is no specific order in which you must rewrite a compound inequality. However, it is often helpful to start by simplifying the individual inequalities and then combining them into a compound inequality.

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