Beyond PSCC Paper: Bound semantics

[frederikgeth]

The "bounded" keyword argument in powermodels may lead to confusion in the stochastic context, where variables are naturally unbounded (for each k) and the bounds are replaced with chance constraints (across all k). We could provide wrapper functions if this ends up being too confusing.

Note: we may also want to explore implied deterministic bounds, given chance-constrained bounds.

[timmyfaraday]

It is inherently a semantics discussion, but in what follows my thoughts on it:

There are two types of variables: i) the random variables and ii) the individual PCE variables, which describe the former.

i) The random variables are bounded by chance constraints, as it is inherently impossible to deterministically bound a continuous random variables; and
ii) the individual PCE variables may be bounded by deterministic constraints. For convergence purposes, I think they should be bounded, but that's a different discussion.

Given that the variable functions, e.g., variable_bus_voltage(..., bounded=bounded), adds the PCE variables to the problem not the random variables, and therefore the bounded argument applies to those variables, I find it consistent with PowerModels.jl to keep this argument as is.

An additional argument could be included to switch on/off the chance constraints on the random variables, e.g., stochastic_bounded.

[timueh]

Pedantic comment from my side: pathological counterexample can be constructed for "i)". Writing something like Prob(x >= 2) >= 0.5" is trivially satisfied in case of x` being uniform. I'd suggest using something like "meaningful chance constraints".

Also, thinking beyond chance constraints, but constraints involving random variables in general, then "i)" and "ii)" may blur, e.g. the expected value of a random variable may be constrained, which can be written exactly in terms of the zero-order PCE coefficient..

but, you know --> pedantic.. ;)