<p>Take <span class="math-container">$\Omega_1=\Omega_2=[0,0.5]$</span>.</p>
<p>Let <span class="math-container">$p(x)=x^4$</span>, so that <span class="math-container">$p''(x)=12x^2&gt;0$</span> on <span class="math-container">$(0,0.5)$</span>.</p>
<p>Let <span class="math-container">$q(x)=0.5x$</span>, which is linear in <span class="math-container">$x$</span>, and <span class="math-container">$K=0.2&gt;0$</span>.</p>
<p>For <span class="math-container">$\bar\gamma=1$</span>, both objective functions attain their respective maximum at <span class="math-container">$x=0.5$</span>. As the following figure shows, objective function <span class="math-container">$(1)$</span> (blue curve) has a higher maximum than that of objective function <span class="math-container">$(2)$</span> (red curve).</p>
<p><a href="https://i.sstatic.net/HsWek.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/HsWek.png" alt="enter image description here" /></a></p>
<p>But for <span class="math-container">$\gamma=5&gt;\bar\gamma$</span>, objective function <span class="math-container">$(1)$</span> attains a maximum value of approximately <span class="math-container">$0.276$</span>, while objective function <span class="math-container">$(2)$</span> attains a higher maximum value of approximately <span class="math-container">$0.309$</span>, contradicting the claim that <span class="math-container">$(1)$</span>'s maximum would always be greater than <span class="math-container">$(2)$</span>'s maximum for <span class="math-container">$\gamma&gt;\bar\gamma$</span>.</p>
<p><a href="https://i.sstatic.net/eHXz6.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/eHXz6.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2020-08-17T04:43:53.603Z
2020-08-17T04:43:53.603Z
Herr K.
<p>Consider the following maximization problems:</p>
<ol>
<li><p><span class="math-container">$\max_{x} x -\gamma p(x)$</span>
subject to <span class="math-container">$x \in \Omega_1$</span></p>
</li>
<li><p><span class="math-container">$\max_{x} x-\gamma (p(x) + q(x) )+K$</span>
subject to <span class="math-container">$x \in \Omega_2$</span></p>
</li>
</ol>
<p>where <span class="math-container">$\Omega_1 $</span> and <span class="math-container">$ \Omega_2$</span> are convex sets, <span class="math-container">$p(x) \geq 0 $</span> and <span class="math-container">$q(x) \geq 0$</span> for all <span class="math-container">$x\in \Omega_2$</span>. Also, <span class="math-container">$p''(x)&gt;0$</span> and <span class="math-container">$q(x)$</span> is linear in <span class="math-container">$x$</span> and <span class="math-container">$K&gt;0$</span> is a constant.</p>
<p>If for given <span class="math-container">$\gamma = \bar{\gamma}$</span>, the optimized objective value for problem 1 was greater than the optimized objective value of problem 2, does the optimized objective value for the problem 1 always greater than that of problem 2 for all <span class="math-container">$\gamma &gt; \bar{\gamma}$</span>?</p>
<p>Prove or provide counter example for (1) <span class="math-container">$\Omega_1= \Omega_2$</span> and (2) <span class="math-container">$\Omega_1 \subset \Omega_2$</span>.</p>
<p>Since the higher penalty proportional to <span class="math-container">$\gamma$</span> is imposed to the objective function of problem 2, this claim seems right. I tried using contradiction, in which assuming there exists <span class="math-container">$\gamma'&gt;\bar{\gamma}$</span> such that optimized value for problem 2 is greater than that of problem 1, but struggling. How can this be proved?</p>

CC BY-SA 4.0
2020-08-15T23:42:58.860Z
What I meqn was optimised value of the objective function, not the optimal $x^*$. I will edit
mike
CC BY-SA 4.0
2020-08-16T18:04:25.493Z
Cross-posted on Operations Research Stack Exchange: https://or.stackexchange.com/q/4681
Flux
CC BY-SA 4.0
2020-08-18T11:13:40.490Z
For $K \leq 0$ the claim is true, but not for $K>0$
Bertrand
<optimization>
How can this be proved? (Convex optimization)
mike
# How can this be proved? (Convex optimization)

(Asked by: username_0 on 2020-08-15)

Consider the following maximization problems:

1. $\max\_{x} x -\gamma p(x)$
   subject to $x \in \Omega\_1$
2. $\max\_{x} x-\gamma (p(x) + q(x) )+K$
   subject to $x \in \Omega\_2$

where $\Omega\_1 $ and $ \Omega\_2$ are convex sets, $p(x) \geq 0 $ and $q(x) \geq 0$ for all $x\in \Omega\_2$. Also, $p''(x)>0$ and $q(x)$ is linear in $x$ and $K>0$ is a constant.

If for given $\gamma = \bar{\gamma}$, the optimized objective value for problem 1 was greater than the optimized objective value of problem 2, does the optimized objective value for the problem 1 always greater than that of problem 2 for all $\gamma > \bar{\gamma}$?

Prove or provide counter example for (1) $\Omega\_1= \Omega\_2$ and (2) $\Omega\_1 \subset \Omega\_2$.

Since the higher penalty proportional to $\gamma$ is imposed to the objective function of problem 2, this claim seems right. I tried using contradiction, in which assuming there exists $\gamma'>\bar{\gamma}$ such that optimized value for problem 2 is greater than that of problem 1, but struggling. How can this be proved?

**Tags:** optimization

> **Comments:**
> **username_0**: What I meqn was optimised value of the objective function, not the optimal $x^*$. I will edit  
> **username_1**: Cross-posted on Operations Research Stack Exchange: https://or.stackexchange.com/q/4681  
> **username_2**: For $K \leq 0$ the claim is true, but not for $K>0$  

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## 1 answers

### Answer by username_3 on 2020-08-17. Score: 2 | Accepted answer

Take $\Omega\_1=\Omega\_2=[0,0.5]$.

Let $p(x)=x^4$, so that $p''(x)=12x^2>0$ on $(0,0.5)$.

Let $q(x)=0.5x$, which is linear in $x$, and $K=0.2>0$.

For $\bar\gamma=1$, both objective functions attain their respective maximum at $x=0.5$. As the following figure shows, objective function $(1)$ (blue curve) has a higher maximum than that of objective function $(2)$ (red curve).

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

But for $\gamma=5>\bar\gamma$, objective function $(1)$ attains a maximum value of approximately $0.276$, while objective function $(2)$ attains a higher maximum value of approximately $0.309$, contradicting the claim that $(1)$'s maximum would always be greater than $(2)$'s maximum for $\gamma>\bar\gamma$.

[![enter image description here](https://i.sstatic.net/eHXz6.png)](https://i.sstatic.net/eHXz6.png)


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