# Veg*n dishes versus constrained optimization

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Mathematical optimization is concerned with  problems of the form $$\text{maximize~} f(x) \text{~ for ~} x \in X$$ for some set $X \subset Y$ and function $f : Y \to \mathbb{R}$. In this post, we’ll think of $Y$ as the set of possible restaurant dishes, $X\subset Y$ as the set of dishes satisfying certain constraints like being digestible, non-poisonous and not containing human flesh. The function $f$ to optimize is some combination of price, healthyness and taste.

A first observation is that for any $X' \subset X$ the maximum of $f(x)$ over $X'$ is no bigger than the maximum over $X$, for if $x \in X'$ attains the maximum of $f(x)$ over $X'$, then also $x \in X$, so the maximum of $f$ over $X$ is at least $f(x)$. In normal words, if you restrict your diet, you can miss out on good dishes, but never gain access to better dishes than on an unrestricted diet.

From this, we could deduce that you should never pick a vegan dish in a restaurant because the non-vegan dishes were made with fewer restrictions and hence can only be better than the vegan dish. Same for choosing recipes to cook yourself. Before I was vegan, this was my conscious reason for always choosing dishes with meat.

But is the deduction true? I don’t think so. Because, unbeknownst to many, the meat dishes are actually constrained to contain meat. I don’t know why, but my two prime suspects are Goodhart’s law impacting the reasoning above, or meat-eaters being scared of vegetables.

As it turns out, making a good vegetarian meal takes non-zero skill, contrary to making a good meal with meat. In my experience, this causes chefs to put actual thought into their vegetarian dishes, causing these to actually be tastier than most dishes with meat. So the argument from mathematical optimization actually gives the wrong answer here!