I am very new to this field, and have very recently started doing some self study on this topic. After reading some papers and reproducing some of the results in them, I am not very clear about what objective exactly is fulfilled by doing portfolio optimization. I assume that having a minimum variance portfolio is of interest or relevance for investors because it tells them objectively which stocks to buy and which ones to sell, is it not? If that is not the case, then could you please explain why would an optimized portfolio be of interest to investors?
Also, if we say that investors should invest in stocks according to the optimized portfolio weights, does it mean that investors should invest in all the stocks anyway – and this method just gives the proportion of each to be invested in? If yes, it is a bit confusing, because in real life, although I myself have not traded, I assume that an investor would typically buy or sell only a small subset of stocks at any given time. Is this not true? So, the more meaningful problem to solve would be, in my mind, to find out which stocks to buy or/and sell at any given time, based on the studied stock returns, is it not?
EDIT: For example, we can use www.portfoliovisualizer.com to check which assets to invest in (according to whatever theory they are using):
The algorithm is not giving as output a portfolio of all the investments that the investor initially had. It is giving a subset of them (I assume the ones most likely to make profit). But portfolio management theory doesn't seem to talk about subsets of assets. Which one is correct, or better?