What confuses me is that if p_i is in the interval [0,1] then usually p_i^h (assuming that is a power we are talking about) is also in that interval. So those bound constraints all reduce to the simple
Again, this assumes that p_i^h is p_i raised to the power h. I suppose that superscript might be defined in any way you like, but if you want us to understand mathematics, then you need to either let us assume that your notation is consistent with standard mathematical notation, or tell us how it is not.
Next, the first equality constraint does not seem to be linear, since it also has powers of the p_i in the sum.
Finally, we don't know what values M and H take on, but there are apparently M unknowns to search over, but we see M+H equality constraints. And that ignores the bound constraints. This is a recipe for a situation with no possible solutions that satisfy the equality constraints.
