Utility theory and portfolio optimization: utility of what exactly?

Question

In finance, a common problem is selection of an optimal portfolio given some constraints (e.g. budget constraint and perhaps nonnegative allocation constraint). One can define the optimization problem based on maximization of expected utility. For example, if the utility function is $u(x)=x-\frac{A}{2}x^2$, then (in absence of constraints aside from the budget constraint) this yields the well-known mean-variance portfolio optimization.

Questions:

1. What is $x$? Is it (portfolio) return (in \$), % or logarithmic return, total wealth (initial wealth + portfolio return) or yet something else?
2. Which choice is typically used in portfolio optimization?
3. Which choice makes most sense from the perspective of utility theory?

I am interested in this in the context of trying to pick a sensible value of $A$ to represent preferences of a "typical" agent. In my experience, $A=2$ as commonly found in the literature does not seem to align well with $x$ being logarithmic return (yields funny portfolios). Thoughts on this will also be appreciated.

Given the popularity of mean-variance portfolio theory and how basic my questions are, I suppose they have been considered before (perhaps even treated in textbooks). References would be appreciated.

Edit: In addition to the answer I got for Question 2, I am still highly interested in obtaining an answer for Question 3 (as well as alternative answers to Question 2).

Answer 1

In mean-variance optimisation I have typically seen the below quadratic utility function where $𝐸[𝑅]$ is the expected return (or mean return) of a possible portfolio, $\sigma^{2}$ is the return variance of that portfolio and $A$ is a parameter that the determines the sensitivity to variance.

$$𝑈=𝐸[𝑅]−\frac{1}{2}A\sigma^{2}$$

As an aside, maximising % return, also maximises $ return, log return and total wealth. They might not be equivalent when optimising for mean and variance though.