Say Bob wants to buy \$30 million worth of APPL stock at a price of \$130.

He decides to use a limit order.

But posting a $30 million limit order would drive the price up and prevent him from being filled.

Obviously he must post only a fraction of this amount and then slowly top it up each time he gets filled. But what fraction? Should he post \$1 million? \$1000? or even \$10?

What variables decide what size or amount of \$ he should post? Surely there must exist an optimal amount, but what is it?