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Your question is little broad and has two aspect: Theory and Application. If you are interested in scientific approach and academic literature this kind of thing is called Mathematical_optimization which is branch of Multi-objective optimization which is again a branch of Operations_research. In terms of mathematics of solving these problems multivariate ...
Yes, a Monte Carlo simulation (MC) is what you need. It is a well known and documented approach with many uses in finance, science and engineering. MC simulations are used to simulate the returns of complex financial assets or in your case returns of business ventures under uncertainty. Your input variables ($x_1, x_2,\cdots, x_n$) are uncertain. If you ...
The easiest way ,i suppose, would be to analyze the market depth. If there is a 20 cent gap between each 100 shares on the bid then to sell 1000 shares instantly would have an impact of $2. Your average price is the midpoint. There are more complicated formulations, but this seems to be how it works on simple examples such as bitcoin exchanges.
I would try a Gradient Descent algorithm if possible, although this might take a little VBA knowledge. In general, finding the absolute minimum of a multidimensional problem like that is complicated and time consuming, but the gradient descent will find a local minimum easily. What you will need to do for worst case scenarios: Construct a cost function ...
You can apply the Kolmogorov-Smirnov test. I simply quote from the entry: "The two-sample K–S test is one of the most useful and general nonparametric methods for comparing two samples, as it is sensitive to differences in both location and shape of the empirical cumulative distribution functions of the two samples." There is an R-implementation too.
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