2-dim tuple for noise

[lorenzoreus]

Dear Oscar,

I'm trying to implement a process for 2 variables in the price interpolation scheme.
Specifically I want to have a noise with a simultaneous but different movement for each variable.
The DiscreteDistribution only allows a vector (N,1) of posibble values for the N possible realizations. But I need to specify a matrix of (N,2) size to include all posibble values. Is it somehow possible to create a noise for DynamicPriceInterpolation in this case?

Thanks!

[odow]

The answer is just to create a vector of the Cartesian product of the discrete random variables.

# Support and probabilities of first random variable:
support_A = [1, 2, 3]
probability_A = [0.5, 0.2, 0.3]

# Support and probabilities of second random variable:
support_B = [5, 6]
probability_B = [0.5, 0.5]

# Vector of joint-supports:
supports = [(a, b) for a in support_A for b in support_B]

# Assuming that the random variables are independent
probabilities = [a * b for a in probability_A for b in probability_B]

# Form a discrete distribution
noise_distribution = DiscreteDistribution(supports, probability)

# Now the argument `noise` in the `dynamics` function will be a tuple `(a, b)`.

[lorenzoreus]

Thanks Oscar!
Following the same approach if I want to add this noise as a tuple. how do operations work?
For example if I have

function modeldynamics(price,noise)
a1=price+noise
return a1
end
HOw price is defined? Is it also a tuple or an array? Is the + adding tuples componentwise?
i) How should I enter min_price? As an array or tuple?
ii) How do I call the component of this price in the SDDPmodel ?
iii) Something that applies for one-dim price process too. Can I write a constraint in terms of the price? I've only seen example where prices are in the @stageobjective with the syntax "price->"

Thanks for everything! I really appreaciate your help.

[odow]

HOw price is defined? Is it also a tuple or an array?

The riverchain example demonstrates N-dimensional prices. In particular

SDDP.jl/examples/PriceInterpolation/riverchain.jl

Lines 68 to 73 in d90faae

	return (t,i) -> DynamicPriceInterpolation(
	dynamics = (p,w) -> ar2price(p, w, t, i),
	initial_price = (pbar, pbar),
	min_price = (minprice, minprice),
	max_price = (maxprice, maxprice),
	noise = NOISES

Is the + adding tuples componentwise?

This isn't defined in Julia. It will error. You have to do it yourself.

i) How should I enter min_price? As an array or tuple?

See first comment.

ii) How do I call the component of this price in the SDDPmodel ?

It will be passed in as a tuple, so access it like a tuple.

SDDP.jl/examples/PriceInterpolation/riverchain.jl

Lines 117 to 121 in d90faae

	@stageobjective(sp,
	price -> sum(
	valley_chain[i].spill_cost * spill[i] for i in 1:N
	) - price[1] * generation_quantity
	)

iii) Something that applies for one-dim price process too. Can I write a constraint in terms of the price? I've only seen example where prices are in the @stageobjective with the syntax "price->"

Unfortunately, this is never allowed. One key assumption that we make in our paper is that price is independent of the states and actions in the model.

If you mean put a constraint in price (e.g., clamp(price, 0, 1)), then this is allowed in some circumstances. It gets a bit tricky, but you need to ensure that concavity of the LP is maintained through this price mapping. In general, constraints such as this clamp destroy that concavity. Caveat emptor. It's up to you if this is a problem. If it isn't you'll get a good heuristic with no guarantee of global optimality.

[lorenzoreus]

Thanks oscar. Very clear explanations.

And one last question related to this issue of price interpolation scheme
Can I somehow keep the current price (not the price at the next stage that includes the noise) as a state variable? I need to use this value, which is known at stage t, in a constraint.

[odow]

I need to use this value, which is known at stage t, in a constraint.

In almost all cases, you can't use the price in a constraint (that is, a JuMP @constraint constraint). The library purposefully excludes this possibility.

If you need to use the price in the prior stage in the price dynamics (for example, an auto-regressive process with lag two), you can augment the price-state dimension so it is a tuple of (price[t], price[t-1]).

[lorenzoreus]

ok! thanks again.