The way that Markowitz (1952) introduces mean variance optimisation to the financial world is as a maths sandwich between two slices of guts. I think in the end, both those pieces of gut will prove amenable to maths too. The first piece of so called guts is Markowitz's 'step one', the idea that one arrives though experience and observation at a set of beliefs (probabilities) concerning future expected performances (general term there, think returns, risks) on a set of risky securities.
For me, this sounds like it was already anticipating Black Litterman, 1990 approach, which was in effect to operationalise experience and observation in a process of Bayesian probabilistic modelling. This approach is itself a form of constrained optimisation, rather like the techniques in linear programming, for example with Lagrange multipliers. The Bayesian approach is of course not limited to linear assumptions.
Prior to 1952, Kantorovich and then Dantzig has produced solutions to linear programming problems. Dantzig, for example, had invented the simplex method when he misinterpreted his professor Jerzy Neyman's list of unsolved problems as a homework exercise, and went ahead and solved it.
So Markowitz goes into this paper knowing there's a solution to his 'step 2', being an optimisation of both mean and variance in a portfolio.
Finally, the second slice of gut involves investors deciding which level of return they want, given their preference for the level of risk they're prepared to bear. I think too, in time, this will be amenable to a mathematical solution. That is to say, their level of risk can, to take only a single example, can become a function of a macro-economic model.
