It can't be difficult to design an auction where the amount of $$ raised is set, with the "price" being the valuation (ie. what % of the company the investors are willing to own).
That’s basically how Treasuries are auctioned. The problem for an individual company is I might be interested to IPO only under certain terms and be willing to pay a middleman to arrange that for the certainty. (The Treasury auction isn’t likely to move much and when it does, well, that’s the price-the government needs the cash to run; that’s not quite the same as a company who has other possibly viable options, including waiting.)
