I am looking for a geometric and intuitive explanation of the conjugate function and how it maps to the below analytical formula.
$$ f^*(y)= \sup_{x \in \operatorname{dom} f } (y^Tx-f(x))$$
Optimization – Geometric Intuition of Conjugate Function
convex optimizationconvex-analysisoptimization
Related Solutions
$$\sup_{\frac{x}{t}\in dom~f\\ t>0}(t(\dfrac{y^Tx}{t}+s-f(\dfrac{x}{t}))) = \sup\{tv \mid v =\sup_{u\in dom~f}((y^Tu+s-f(u)))\text{ and }\color{red}{t > 0}\}$$
Edit to address a question in the comments
What does $$\sup_{\frac{x}{t}\in dom~f\\ t>0}(t(\dfrac{y^Tx}{t}+s-f(\dfrac{x}{t})))$$ mean?
It means we form a set $$A =\left\{t(\dfrac{y^Tx}{t}+s-f(\dfrac{x}{t}))~ \right|~\left. t > 0, \frac xt \in dom~f\right\}$$ Then we find its supremum.
But that is a bit ungainly. After all, everywhere that $x$ occurs in the definition of $A$, it is as a part of $x/t$. So why not just use that value directly instead of expressing it in terms of this other arbitrary value $x$? So I replaced $x/t$ with $u$ to get $$A = \left\{t(y^Tu+s-f(u))\mid t > 0, u \in dom~f\right\}$$
For the next step, I just made use of this equivalence:
$$\sup\{\phi(r,s)\mid r\in A, s\in B\} = \sup\{\sup\{\phi(r,s)\mid s\in B\}\mid r \in A\}$$
where $\phi : A\times B \to \Bbb R$. This is easy enough to see: For any value $p$ in the LH set, $p = \phi(r,s)$ for some $r,s$, and so $p\le \sup\{\phi(r,s)\mid s\in B\} \le \sup\{\sup\{\phi(r,s)\mid s\in B\}\mid r \in A\}$. Thus the LH side is $\le$ the RH side. Conversely, For any value $h$ less than the RH side, there must be an $r \in A$ with $\sup\{\phi(r,s)\mid s\in B\} > h$, which in turn means there is an $s \in B$ with $\phi(r,s) > h$, which means that the LH side is also $> h$. Hence they must be equal.
Of course this leaves you with a problem, which wasn't apparent in my original answer, because I wrote down the wrong thing. You will note that originally I had $tv \in dom~f$. I had meant this to be $t > 0$ (which correction is now highlighted in red, if your browser shows colored mathjax) but in my various editings of the form, I had accidently left in the wrong thing and deleted the correct one.
But in this form, it is quite clear. Since $t$ can be arbirarily large, the common value of these two expressions is $\infty$ (assuming that $f^*$ has at least one positive value). Once it is made obvious by my version, you can check that it holds in your version as well. Pick an arbitrary $u \in dom~f$ with $f(u) > 0$ and for any $t > 0$, you can take $x = tu$ to make the supremum as large as you like.
There are two possibilities here. Either the point of this exercise is for you to discover the complex conjugate of a perspective function is always infinite, or else, by conjugate of the perspective function, they only mean to conjugate with respect to constant $t$:
$$ g^*(y,t)=\sup_{\frac{x}{t}\in dom~f}(t(\dfrac{y^Tx}{t}+s-f(\dfrac{x}{t})))$$
I can't say which is the case for your course.
For any given $x \in \operatorname{dom} f$, the function: $$y \mapsto y^\top x - f(x)$$ is an affine function, which is convex and lower semicontinuous (and, indeed, continuous). The epigraph of these functions is therefore closed and convex.
Then, $f^*$ is the pointwise supremum of the above functions, and its epigraph is the intersection of the above affine functions' epigraphs. Each are closed and convex, proving the epigraph of $f^*$ is closed and convex, thus $f^*$ is convex and lower semicontinuous.
Best Answer
To me the best interpretation is economic. Interpret $f(x)$ as the cost to produce the quantity $x$ of some product and interpret $y$ as the market price per unit. It is easy to see that $f^*(y)$ represents the optimal profit at given prices $y$. The quantity $xy$ represents revenue from sales and $f(x)$ represents production costs.
Now for the geometrical interpretation. If you sketch the graph of the costs of production $f(x)$ and assume it convex, continuous, and differentiable, you will see that the point of optimal production, given prices $y$, is given by $y - f'(x)=0$, and this can be found graphically with a ruler, looking for the tangent in the cost curve with the same slope $y$. If you place the ruler in that tangent point, it can be seen that the ruler intersection with the vertical axis will give $-(xy - f(x))$.
This is a very useful calculating device. Provided only with the graph of $f(x)$ and a ruler, the analyst is able to turn the ruler and find what is the optimal profit for each possible price. This can be plotted into another piece of paper. Then given any price $y$ he is able to find what was the optimal profit. Without noticing, he has discovered the conjugate function.