<p>Real-valued Monotonic functions defined on real line or subset of real line are both quasi-concave and quasi-convex, but that is not necessarily the case if the function is defined on <span class="math-container">$\mathbb{R}^n$</span> or its subset, where <span class="math-container">$n\geq 2$</span>. For example, all these are monotonic functions:</p>
<ul>
<li><span class="math-container">$f$</span> defined on <span class="math-container">$\mathbb{R}^2_+$</span> and as <span class="math-container">$f(x_1,x_2)=x_1^\frac{1}{2}x_2^\frac{1}{2}$</span> is quasi-concave, but not quasi-convex.</li>
<li><span class="math-container">$f$</span> defined on <span class="math-container">$\mathbb{R}^2_+$</span> and as <span class="math-container">$f(x_1,x_2)=x_1^2+x_2^2$</span> is quasi-convex, but not quasi-concave.</li>
<li><span class="math-container">$f$</span> defined on <span class="math-container">$\mathbb{R}^2_+$</span> and as <span class="math-container">$f(x_1,x_2)=x_1 + x_2$</span> is both quasi-convex and quasi-concave.</li>
<li><span class="math-container">$f$</span> defined on <span class="math-container">$\mathbb{R}^2_+$</span> and as <span class="math-container">$f(x_1,x_2)=\sqrt{x_1} + x_2^2$</span> is neither quasi-convex nor quasi-concave.</li>
</ul>

CC BY-SA 4.0
2023-02-14T17:13:17.740Z
Superb! So it actually was option 1 regarding the possible places I thought I might be going wrong! Would you care to distinguish, monotonic, and strictly increasing for me?
CormJack
CC BY-SA 4.0
2023-02-14T17:11:42.177Z
2023-02-14T17:11:42.177Z
Amit
<p>I believe i have a major misunderstanding surrounding quasi-convex constraints in maximisation, when using monotone functions.  Can you help me spot my errors please?</p>
<p>The definition of a quasi-convex function that I will be using is:</p>
<p>A function <span class="math-container">$f$</span> from <span class="math-container">$\mathbb{R^n}$</span> to <span class="math-container">$\mathbb{R}$</span>  is <em>quasi-convex</em> if, <span class="math-container">$\forall \alpha \in \mathbb{R^n}$</span>, the set</p>
<p><span class="math-container">$$\{x\in D: f(x)&lt;\alpha\},$$</span><br />
where <span class="math-container">$D$</span> is the domain of <span class="math-container">$f$</span>, is a <em>convex</em> set.</p>
<p><strong>Monotonic Functions are Quasi-convex and Quasi-concave</strong> - <a href="https://en.wikipedia.org/wiki/Quasiconvex_function" rel="nofollow noreferrer">Wikipedia</a></p>
<ol>
<li>If i take <span class="math-container">$f: \mathbb{R^{2+}} \to \mathbb{R^+}$</span> where <span class="math-container">$f=(x_1x_2)^{1/2}$</span> why are my lower contour sets not convex? (See attached image). In my mind, the function is monotonic, therefore it is quasi-convex and quasi-concave, but i cannot show quasi convexity using the contours, as per the definition.</li>
</ol>
<p><a href="https://i.sstatic.net/H5ex6.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/H5ex6.png" alt="enter image description here" /></a></p>
<p><strong>Quasi-concave constraint in minimisation problem</strong></p>
<ol>
<li>A lecture video <a href="https://youtu.be/9tUj530Mw_Q?t=674" rel="nofollow noreferrer">times-tamped</a> I was watching made the point that when defining our equality constraint in a minimisation problem, we have to write <span class="math-container">$f(x_1, x_2) - q = 0$</span>, rather than <span class="math-container">$q - f(x_1, x_2) = 0$</span> because we need the constraint to be quasi-concave. But again in the video i believe the constraint is strictly increasing, so surely the constraint is already quasi-convex and quasi-concave? So it doesn't matter?</li>
</ol>
<p><strong>Where I am possibly going wrong</strong></p>
<ol>
<li>I'm misunderstanding the relationship between a monotonic function and quasi-convexity/concavity? Perhaps its different when our domain is in <span class="math-container">$\mathbb{R^n}$</span> vs <span class="math-container">$\mathbb{R}$</span></li>
<li>I'm misunderstanding the terms monotonic, strictly increasing, and non decreasing, and they apply differently to quasi-convexity/concavity than I imagine?</li>
<li>Everyone else is wrong and I'm about to reinvent optimisation theory?</li>
</ol>

<utility><optimization><cost-functions>
Quasi-convex constraints using monotonic functions
CormJack
# Quasi-convex constraints using monotonic functions

(Asked by: username_0 on 2023-02-14)

I believe i have a major misunderstanding surrounding quasi-convex constraints in maximisation, when using monotone functions. Can you help me spot my errors please?

The definition of a quasi-convex function that I will be using is:

A function $f$ from $\mathbb{R^n}$ to $\mathbb{R}$ is *quasi-convex* if, $\forall \alpha \in \mathbb{R^n}$, the set

$$\{x\in D: f(x)<\alpha\},$$  
where $D$ is the domain of $f$, is a *convex* set.

**Monotonic Functions are Quasi-convex and Quasi-concave** - [Wikipedia](https://en.wikipedia.org/wiki/Quasiconvex_function)

1. If i take $f: \mathbb{R^{2+}} \to \mathbb{R^+}$ where $f=(x\_1x\_2)^{1/2}$ why are my lower contour sets not convex? (See attached image). In my mind, the function is monotonic, therefore it is quasi-convex and quasi-concave, but i cannot show quasi convexity using the contours, as per the definition.

[![enter image description here](https://i.sstatic.net/H5ex6.png)](https://i.sstatic.net/H5ex6.png)

**Quasi-concave constraint in minimisation problem**

1. A lecture video [times-tamped](https://youtu.be/9tUj530Mw_Q?t=674) I was watching made the point that when defining our equality constraint in a minimisation problem, we have to write $f(x\_1, x\_2) - q = 0$, rather than $q - f(x\_1, x\_2) = 0$ because we need the constraint to be quasi-concave. But again in the video i believe the constraint is strictly increasing, so surely the constraint is already quasi-convex and quasi-concave? So it doesn't matter?

**Where I am possibly going wrong**

1. I'm misunderstanding the relationship between a monotonic function and quasi-convexity/concavity? Perhaps its different when our domain is in $\mathbb{R^n}$ vs $\mathbb{R}$
2. I'm misunderstanding the terms monotonic, strictly increasing, and non decreasing, and they apply differently to quasi-convexity/concavity than I imagine?
3. Everyone else is wrong and I'm about to reinvent optimisation theory?

**Tags:** utility, optimization, cost-functions

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## 1 answers

### Answer by username_1 on 2023-02-14. Score: 3 | Accepted answer

Real-valued Monotonic functions defined on real line or subset of real line are both quasi-concave and quasi-convex, but that is not necessarily the case if the function is defined on $\mathbb{R}^n$ or its subset, where $n\geq 2$. For example, all these are monotonic functions:

* $f$ defined on $\mathbb{R}^2\_+$ and as $f(x\_1,x\_2)=x\_1^\frac{1}{2}x\_2^\frac{1}{2}$ is quasi-concave, but not quasi-convex.
* $f$ defined on $\mathbb{R}^2\_+$ and as $f(x\_1,x\_2)=x\_1^2+x\_2^2$ is quasi-convex, but not quasi-concave.
* $f$ defined on $\mathbb{R}^2\_+$ and as $f(x\_1,x\_2)=x\_1 + x\_2$ is both quasi-convex and quasi-concave.
* $f$ defined on $\mathbb{R}^2\_+$ and as $f(x\_1,x\_2)=\sqrt{x\_1} + x\_2^2$ is neither quasi-convex nor quasi-concave.

> **Comments:**
> **username_0**: Superb! So it actually was option 1 regarding the possible places I thought I might be going wrong! Would you care to distinguish, monotonic, and strictly increasing for me?  

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