Risk aversion

This tutorial was generated using Literate.jl. Download the source as a .jl file. Download the source as a .ipynb file.

In Introductory theory, we implemented a basic version of the SDDP algorithm. This tutorial extends that implementation to add risk-aversion.

Packages

This tutorial uses the following packages. For clarity, we call import PackageName so that we must prefix PackageName. to all functions and structs provided by that package. Everything not prefixed is either part of base Julia, or we wrote it.

import ForwardDiff
import HiGHS
import Ipopt
import JuMP
import Statistics

Risk aversion: what and why?

Often, the agents making decisions in complex systems are risk-averse, that is, they care more about avoiding very bad outcomes, than they do about having a good average outcome.

As an example, consumers in a hydro-thermal problem may be willing to pay a slightly higher electricity price on average, if it means that there is a lower probability of blackouts.

Risk aversion in multistage stochastic programming has been well studied in the academic literature, and is widely used in production implementations around the world.

Risk measures

One way to add risk aversion to models is to use a risk measure. A risk measure is a function that maps a random variable to a real number.

You are probably already familiar with lots of different risk measures. For example, the mean, median, mode, and maximum are all risk measures.

We call the act of applying a risk measure to a random variable "computing the risk" of a random variable.

To keep things simple, and because we need it for SDDP, we restrict our attention to random variables $Z$ with a finite sample space $\Omega$ and positive probabilities $p_{\omega}$ for all $\omega \in \Omega$. We denote the realizations of $Z$ by $Z(\omega) = z_{\omega}$.

A risk measure, $\mathbb{F}[Z]$, is a convex risk measure if it satisfies the following axioms:

Axiom 1: monotonicity

Given two random variables $Z_1$ and $Z_2$, with $Z_1 \le Z_2$ almost surely, then $\mathbb{F}[Z_1] \le F[Z_2]$.

Axiom 2: translation equivariance

Given two random variables $Z_1$ and $Z_2$, then for all $a \in \mathbb{R}$, $\mathbb{F}[Z + a] = \mathbb{F}[Z] + a$.

Axiom 3: convexity

Given two random variables $Z_1$ and $Z_2$, then for all $a \in [0, 1]$,

\[\mathbb{F}[a Z_1 + (1 - a) Z_2] \le a \mathbb{F}[Z_1] + (1-a)\mathbb{F}[Z_2].\]

Now we know what a risk measure is, let's see how we can use them to form risk-averse decision rules.

Risk-averse decision rules: Part I

We started this tutorial by explaining that we are interested in risk aversion because some agents are risk-averse. What that really means, is that they want a policy that is also risk-averse. The question then becomes, how do we create risk-averse decision rules and policies?

Recall from Introductory theory that we can form an optimal decision rule using the recursive formulation:

\[\begin{aligned} V_i(x, \omega) = \min\limits_{\bar{x}, x^\prime, u} \;\; & C_i(\bar{x}, u, \omega) + \mathbb{E}_{j \in i^+, \varphi \in \Omega_j}[V_j(x^\prime, \varphi)]\\ & x^\prime = T_i(\bar{x}, u, \omega) \\ & u \in U_i(\bar{x}, \omega) \\ & \bar{x} = x, \end{aligned}\]

where our decision rule, $\pi_i(x, \omega)$, solves this optimization problem and returns a $u^*$ corresponding to an optimal solution.

If we can replace the expectation operator $\mathbb{E}$ with another (more risk-averse) risk measure $\mathbb{F}$, then our decision rule will attempt to choose a control decision now that minimizes the risk of the future costs, as opposed to the expectation of the future costs. This makes our decisions more risk-averse, because we care more about the worst outcomes than we do about the average.

Therefore, we can form a risk-averse decision rule using the formulation:

\[\begin{aligned} V_i(x, \omega) = \min\limits_{\bar{x}, x^\prime, u} \;\; & C_i(\bar{x}, u, \omega) + \mathbb{F}_{j \in i^+, \varphi \in \Omega_j}[V_j(x^\prime, \varphi)]\\ & x^\prime = T_i(\bar{x}, u, \omega) \\ & u \in U_i(\bar{x}, \omega) \\ & \bar{x} = x. \end{aligned}\]

To convert this problem into a tractable equivalent, we apply Kelley's algorithm to the risk-averse cost-to-go term $\mathbb{F}_{j \in i^+, \varphi \in \Omega_j}[V_j(x^\prime, \varphi)]$, to obtain the approximated problem:

\[\begin{aligned} V_i^K(x, \omega) = \min\limits_{\bar{x}, x^\prime, u} \;\; & C_i(\bar{x}, u, \omega) + \theta\\ & x^\prime = T_i(\bar{x}, u, \omega) \\ & u \in U_i(\bar{x}, \omega) \\ & \bar{x} = x \\ & \theta \ge \mathbb{F}_{j \in i^+, \varphi \in \Omega_j}\left[V_j^k(x^\prime_k, \varphi)\right] + \frac{d}{dx^\prime}\mathbb{F}_{j \in i^+, \varphi \in \Omega_j}\left[V_j^k(x^\prime_k, \varphi)\right]^\top (x^\prime - x^\prime_k)\quad k=1,\ldots,K. \end{aligned}\]

Warning

Note how we need to explicitly compute a risk-averse subgradient! (We need a subgradient because the function might not be differentiable.) When constructing cuts with the expectation operator in Introductory theory, we implicitly used the law of total expectation to combine the two expectations; we can't do that for a general risk measure.

Homework challenge

If it's not obvious why we can use Kelley's here, try to use the axioms of a convex risk measure to show that $\mathbb{F}_{j \in i^+, \varphi \in \Omega_j}[V_j(x^\prime, \varphi)]$ is a convex function w.r.t. $x^\prime$ if $V_j$ is also a convex function.

Our challenge is now to find a way to compute the risk-averse cost-to-go function $\mathbb{F}_{j \in i^+, \varphi \in \Omega_j}\left[V_j^k(x^\prime_k, \varphi)\right]$, and a way to compute a subgradient of the risk-averse cost-to-go function with respect to $x^\prime$.

Primal risk measures

Now we know what a risk measure is, and how we will use it, let's implement some code to see how we can compute the risk of some random variables.

Note

We're going to start by implementing the primal version of each risk measure. We implement the dual version in the next section.

First, we need some data:

Z = [1.0, 2.0, 3.0, 4.0]
4-element Vector{Float64}:
 1.0
 2.0
 3.0
 4.0

with probabilities:

p = [0.1, 0.2, 0.4, 0.3]
4-element Vector{Float64}:
 0.1
 0.2
 0.4
 0.3

We're going to implement a number of different risk measures, so to leverage Julia's multiple dispatch, we create an abstract type:

abstract type AbstractRiskMeasure end

and function to overload:

"""
    primal_risk(F::AbstractRiskMeasure, Z::Vector{<:Real}, p::Vector{Float64})

Use `F` to compute the risk of the random variable defined by a vector of costs
`Z` and non-zero probabilities `p`.
"""
function primal_risk end
Main.primal_risk
Note

We want Vector{<:Real} instead of Vector{Float64} because we're going to automatically differentiate this function in the next section.

Expectation

The expectation, $\mathbb{E}$, also called the mean or the average, is the most widely used convex risk measure. The expectation of a random variable is just the sum of $Z$ weighted by the probability:

\[\mathbb{F}[Z] = \mathbb{E}_p[Z] = \sum\limits_{\omega\in\Omega} p_{\omega} z_{\omega}.\]

struct Expectation <: AbstractRiskMeasure end

function primal_risk(::Expectation, Z::Vector{<:Real}, p::Vector{Float64})
    return sum(p[i] * Z[i] for i in 1:length(p))
end
primal_risk (generic function with 1 method)

Let's try it out:

primal_risk(Expectation(), Z, p)
2.9000000000000004

WorstCase

The worst-case risk measure, also called the maximum, is another widely used convex risk measure. This risk measure doesn't care about the probability vector p, only the cost vector Z:

\[\mathbb{F}[Z] = \max[Z] = \max\limits_{\omega\in\Omega} z_{\omega}.\]

struct WorstCase <: AbstractRiskMeasure end

function primal_risk(::WorstCase, Z::Vector{<:Real}, ::Vector{Float64})
    return maximum(Z)
end
primal_risk (generic function with 2 methods)

Let's try it out:

primal_risk(WorstCase(), Z, p)
4.0

Entropic

A more interesting, and less widely used risk measure is the entropic risk measure. The entropic risk measure is parameterized by a value $\gamma > 0$, and computes the risk of a random variable as:

\[\mathbb{F}_\gamma[Z] = \frac{1}{\gamma}\log\left(\mathbb{E}_p[e^{\gamma Z}]\right) = \frac{1}{\gamma}\log\left(\sum\limits_{\omega\in\Omega}p_{\omega} e^{\gamma z_{\omega}}\right).\]

Homework challenge

Prove that the entropic risk measure satisfies the three axioms of a convex risk measure.

struct Entropic <: AbstractRiskMeasure
    γ::Float64
    function Entropic(γ)
        if !(γ > 0)
            throw(DomainError(γ, "Entropic risk measure must have γ > 0."))
        end
        return new(γ)
    end
end

function primal_risk(F::Entropic, Z::Vector{<:Real}, p::Vector{Float64})
    return 1 / F.γ * log(sum(p[i] * exp(F.γ * Z[i]) for i in 1:length(p)))
end
primal_risk (generic function with 3 methods)
Warning

exp(x) overflows when $x > 709$. Therefore, if we are passed a vector of Float64, use arbitrary precision arithmetic with big.(Z).

function primal_risk(F::Entropic, Z::Vector{Float64}, p::Vector{Float64})
    return Float64(primal_risk(F, big.(Z), p))
end
primal_risk (generic function with 4 methods)

Let's try it out for different values of $\gamma$:

for γ in [0.001, 0.01, 0.1, 1.0, 10.0, 100.0, 1_000.0]
    println("γ = $(γ), F[Z] = ", primal_risk(Entropic(γ), Z, p))
end
γ = 0.001, F[Z] = 2.9004449279791005
γ = 0.01, F[Z] = 2.9044427792027596
γ = 0.1, F[Z] = 2.9437604953310674
γ = 1.0, F[Z] = 3.264357634151263
γ = 10.0, F[Z] = 3.879608772845574
γ = 100.0, F[Z] = 3.987960271956741
γ = 1000.0, F[Z] = 3.998796027195674
Info

The entropic has two extremes. As $\gamma \rightarrow 0$, the entropic acts like the expectation risk measure, and as $\gamma \rightarrow \infty$, the entropic acts like the worst-case risk measure.

Computing risk measures this way works well for computing the primal value. However, there isn't an obvious way to compute a subgradient of the risk-averse cost-to-go function, which we need for our cut calculation.

There is a nice solution to this problem, and that is to use the dual representation of a risk measure, instead of the primal.

Dual risk measures

Convex risk measures have a dual representation as follows:

\[\mathbb{F}[Z] = \sup\limits_{q \in\mathcal{M}(p)} \mathbb{E}_q[Z] - \alpha(p, q),\]

where $\alpha$ is a concave function that maps the probability vectors $p$ and $q$ to a real number, and $\mathcal{M}(p) \subseteq \mathcal{P}$ is a convex subset of the probability simplex:

\[\mathcal{P} = \{p \ge 0\;|\;\sum\limits_{\omega\in\Omega}p_{\omega} = 1\}.\]

The dual of a convex risk measure can be interpreted as taking the expectation of the random variable $Z$ with respect to the worst probability vector $q$ that lies within the set $\mathcal{M}$, less some concave penalty term $\alpha(p, q)$.

If we define a function dual_risk_inner that computes q and α:

"""
    dual_risk_inner(
        F::AbstractRiskMeasure, Z::Vector{Float64}, p::Vector{Float64}
    )::Tuple{Vector{Float64},Float64}

Return a tuple formed by the worst-case probability vector `q` and the
corresponding evaluation `α(p, q)`.
"""
function dual_risk_inner end
Main.dual_risk_inner

then we can write a generic dual_risk function as:

function dual_risk(
    F::AbstractRiskMeasure,
    Z::Vector{Float64},
    p::Vector{Float64},
)
    q, α = dual_risk_inner(F, Z, p)
    return sum(q[i] * Z[i] for i in 1:length(q)) - α
end
dual_risk (generic function with 1 method)

Expectation

For the expectation risk measure, $\mathcal{M}(p) = \{p\}$, and $\alpha(\cdot, \cdot) = 0$. Therefore:

function dual_risk_inner(::Expectation, ::Vector{Float64}, p::Vector{Float64})
    return p, 0.0
end
dual_risk_inner (generic function with 1 method)

We can check we get the same result as the primal version:

dual_risk(Expectation(), Z, p) == primal_risk(Expectation(), Z, p)
true

Worst-case

For the worst-case risk measure, $\mathcal{M}(p) = \mathcal{P}$, and $\alpha(\cdot, \cdot) = 0$. Therefore, the dual representation just puts all of the probability weight on the maximum outcome:

function dual_risk_inner(::WorstCase, Z::Vector{Float64}, ::Vector{Float64})
    q = zeros(length(Z))
    _, index = findmax(Z)
    q[index] = 1.0
    return q, 0.0
end
dual_risk_inner (generic function with 2 methods)

We can check we get the same result as the primal version:

dual_risk(WorstCase(), Z, p) == primal_risk(WorstCase(), Z, p)
true

Entropic

For the entropic risk measure, $\mathcal{M}(p) = \mathcal{P}$, and:

\[\alpha(p, q) = \frac{1}{\gamma}\sum\limits_{\omega\in\Omega} q_\omega \log\left(\frac{q_\omega}{p_{\omega}}\right).\]

One way to solve the dual problem is to explicitly solve a nonlinear optimization problem:

function dual_risk_inner(F::Entropic, Z::Vector{Float64}, p::Vector{Float64})
    N = length(p)
    model = JuMP.Model(Ipopt.Optimizer)
    JuMP.set_silent(model)
    # For this problem, the solve is more accurate if we turn off problem
    # scaling.
    JuMP.set_optimizer_attribute(model, "nlp_scaling_method", "none")
    JuMP.@variable(model, 0 <= q[1:N] <= 1)
    JuMP.@constraint(model, sum(q) == 1)
    JuMP.@NLexpression(
        model,
        α,
        1 / F.γ * sum(q[i] * log(q[i] / p[i]) for i in 1:N),
    )
    JuMP.@NLobjective(model, Max, sum(q[i] * Z[i] for i in 1:N) - α)
    JuMP.optimize!(model)
    return JuMP.value.(q), JuMP.value(α)
end
dual_risk_inner (generic function with 3 methods)

We can check we get the same result as the primal version:

for γ in [0.001, 0.01, 0.1, 1.0, 10.0, 100.0]
    primal = primal_risk(Entropic(γ), Z, p)
    dual = dual_risk(Entropic(γ), Z, p)
    success = primal ≈ dual ? "✓" : "×"
    println("$(success) γ = $(γ), primal = $(primal), dual = $(dual)")
end
✓ γ = 0.001, primal = 2.9004449279791005, dual = 2.900444927979128
✓ γ = 0.01, primal = 2.9044427792027596, dual = 2.9044427792027667
✓ γ = 0.1, primal = 2.9437604953310674, dual = 2.9437604953310674
✓ γ = 1.0, primal = 3.264357634151263, dual = 3.264357634151263
✓ γ = 10.0, primal = 3.879608772845574, dual = 3.8796087723675954
✓ γ = 100.0, primal = 3.987960271956741, dual = 3.987960271956741
Info

This method of solving the dual problem "on-the-side" is used by SDDP.jl for a number of risk measures, including a distributionally robust risk measure with the Wasserstein distance. Check out all the risk measures that SDDP.jl supports in Add a risk measure.

The "on-the-side" method is very general, and it lets us incorporate any convex risk measure into SDDP. However, this comes at an increased computational cost and potential numerical issues (e.g., not converging to the exact solution).

However, for the entropic risk measure, Dowson, Morton, and Pagnoncelli (2020) derive the following closed form solution for $q^*$:

\[q_\omega^* = \frac{p_{\omega} e^{\gamma z_{\omega}}}{\sum\limits_{\varphi \in \Omega} p_{\varphi} e^{\gamma z_{\varphi}}}.\]

This is faster because we don't need to use Ipopt, and it avoids some of the numerical issues associated with solving a nonlinear program.

function dual_risk_inner(F::Entropic, Z::Vector{Float64}, p::Vector{Float64})
    q, α = zeros(length(p)), big(0.0)
    peγz = p .* exp.(F.γ .* big.(Z))
    sum_peγz = sum(peγz)
    for i in 1:length(q)
        big_q = peγz[i] / sum_peγz
        α += big_q * log(big_q / p[i])
        q[i] = Float64(big_q)
    end
    return q, Float64(α / F.γ)
end
dual_risk_inner (generic function with 3 methods)
Warning

Again, note that we use big to avoid introducing overflow errors, before explicitly casting back to Float64 for the values we return.

We can check we get the same result as the primal version:

for γ in [0.001, 0.01, 0.1, 1.0, 10.0, 100.0]
    primal = primal_risk(Entropic(γ), Z, p)
    dual = dual_risk(Entropic(γ), Z, p)
    success = primal ≈ dual ? "✓" : "×"
    println("$(success) γ = $(γ), primal = $(primal), dual = $(dual)")
end
✓ γ = 0.001, primal = 2.9004449279791005, dual = 2.9004449279791005
✓ γ = 0.01, primal = 2.9044427792027596, dual = 2.9044427792027596
✓ γ = 0.1, primal = 2.9437604953310674, dual = 2.943760495331067
✓ γ = 1.0, primal = 3.264357634151263, dual = 3.264357634151263
✓ γ = 10.0, primal = 3.879608772845574, dual = 3.879608772845574
✓ γ = 100.0, primal = 3.987960271956741, dual = 3.987960271956741

Risk-averse subgradients

We ended the section on primal risk measures by explaining how we couldn't use the primal risk measure in the cut calculation because we needed some way of computing a risk-averse subgradient:

\[\theta \ge \mathbb{F}_{j \in i^+, \varphi \in \Omega_j}\left[V_j^k(x^\prime_k, \varphi)\right] + \frac{d}{dx^\prime}\mathbb{F}_{j \in i^+, \varphi \in \Omega_j}\left[V_j^k(x^\prime_k, \varphi)\right]^\top (x^\prime - x^\prime_k).\]

The reason we use the dual representation is because of the following theorem, which explains how to compute a risk-averse gradient.

The risk-averse subgradient theorem

Let $\omega \in \Omega$ index a random vector with finite support and with nominal probability mass function, $p \in \mathcal{P}$, which satisfies $p > 0$.

Consider a convex risk measure, $\mathbb{F}$, with a convex risk set, $\mathcal{M}(p)$, so that $\mathbb{F}$ can be expressed as the dual form.

Let $V(x,\omega)$ be convex with respect to $x$ for all fixed $\omega\in\Omega$, and let $\lambda(\tilde{x}, \omega)$ be a subgradient of $V(x,\omega)$ with respect to $x$ at $x = \tilde{x}$ for each $\omega \in \Omega$.

Then, $\sum_{\omega\in\Omega}q^*_{\omega} \lambda(\tilde{x},\omega)$ is a subgradient of $\mathbb{F}[V(x,\omega)]$ at $\tilde{x}$, where

\[q^* \in \argmax_{q \in \mathcal{M}(p)}\left\{{\mathbb{E}}_q[V(\tilde{x},\omega)] - \alpha(p, q)\right\}.\]

This theorem can be a little hard to unpack, so let's see an example:

function dual_risk_averse_subgradient(
    V::Function,
    # Use automatic differentiation to compute the gradient of V w.r.t. x,
    # given a fixed ω.
    λ::Function = (x, ω) -> ForwardDiff.gradient(x -> V(x, ω), x);
    F::AbstractRiskMeasure,
    Ω::Vector,
    p::Vector{Float64},
    x̃::Vector{Float64},
)
    # Evaluate the function at x=x̃ for all ω ∈ Ω.
    V_ω = [V(x̃, ω) for ω in Ω]
    # Solve the dual problem to obtain an optimal q^*.
    q, α = dual_risk_inner(F, V_ω, p)
    # Compute the risk-averse subgradient by taking the expectation of the
    # subgradients w.r.t. q^*.
    dVdx = sum(q[i] * λ(x̃, ω) for (i, ω) in enumerate(Ω))
    return dVdx
end
dual_risk_averse_subgradient (generic function with 2 methods)

We can compare the subgradient obtained with the dual form against the automatic differentiation of the primal_risk function.

function primal_risk_averse_subgradient(
    V::Function;
    F::AbstractRiskMeasure,
    Ω::Vector,
    p::Vector{Float64},
    x̃::Vector{Float64},
)
    inner(x) = primal_risk(F, [V(x, ω) for ω in Ω], p)
    return ForwardDiff.gradient(inner, x̃)
end
primal_risk_averse_subgradient (generic function with 1 method)

As our example function, we use:

V(x, ω) = ω * x[1]^2
V (generic function with 1 method)

with:

Ω = [1.0, 2.0, 3.0]
3-element Vector{Float64}:
 1.0
 2.0
 3.0

and:

p = [0.3, 0.4, 0.3]
3-element Vector{Float64}:
 0.3
 0.4
 0.3

at the point:

x̃ = [3.0]
1-element Vector{Float64}:
 3.0

If $\mathbb{F}$ is the expectation risk-measure, then:

\[\mathbb{F}[V(x, \omega)] = 2 x^2.\]

The function evaluation $x=3$ is $18$ and the subgradient is $12$. Let's check we get it right with the dual form:

dual_risk_averse_subgradient(V; F = Expectation(), Ω = Ω, p = p, x̃ = x̃)
1-element Vector{Float64}:
 12.0

and the primal form:

primal_risk_averse_subgradient(V; F = Expectation(), Ω = Ω, p = p, x̃ = x̃)
1-element Vector{Float64}:
 12.0

If $\mathbb{F}$ is the worst-case risk measure, then:

\[\mathbb{F}[V(x, \omega)] = 3 x^2.\]

The function evaluation at $x=3$ is $27$, and the subgradient is $18$. Let's check we get it right with the dual form:

dual_risk_averse_subgradient(V; F = WorstCase(), Ω = Ω, p = p, x̃ = x̃)
1-element Vector{Float64}:
 18.0

and the primal form:

primal_risk_averse_subgradient(V; F = WorstCase(), Ω = Ω, p = p, x̃ = x̃)
1-element Vector{Float64}:
 18.0

If $\mathbb{F}$ is the entropic risk measure, the math is a little more difficult to derive analytically. However, we can check against our primal version:

for γ in [0.001, 0.01, 0.1, 1.0, 10.0, 100.0]
    dual =
        dual_risk_averse_subgradient(V; F = Entropic(γ), Ω = Ω, p = p, x̃ = x̃)
    primal = primal_risk_averse_subgradient(
        V;
        F = Entropic(γ),
        Ω = Ω,
        p = p,
        x̃ = x̃,
    )
    success = primal ≈ dual ? "✓" : "×"
    println("$(success) γ = $(γ), primal = $(primal), dual = $(dual)")
end
✓ γ = 0.001, primal = [12.03239965008496], dual = [12.03239965008496]
✓ γ = 0.01, primal = [12.323650575272044], dual = [12.323650575272044]
✓ γ = 0.1, primal = [14.9332498638561], dual = [14.933249863856101]
✓ γ = 1.0, primal = [17.999012701279042], dual = [17.999012701279042]
✓ γ = 10.0, primal = [17.999999999999996], dual = [18.0]
× γ = 100.0, primal = [NaN], dual = [18.0]

Uh oh! What happened with the last line? It looks our primal_risk_averse_subgradient encountered an error and returned a subgradient of NaN. This is because of the overflow issue with exp(x). However, we can be confident that our dual method of computing the risk-averse subgradient is both correct and more numerically robust than the primal version.

Info

As another sanity check, notice how as $\gamma \rightarrow 0$, we tend toward the solution of the expectation risk-measure [12], and as $\gamma \rightarrow \infty$, we tend toward the solution of the worse-case risk measure [18].

Risk-averse decision rules: Part II

Why is the risk-averse subgradient theorem helpful? Using the dual representation of a convex risk measure, we can re-write the cut:

\[\theta \ge \mathbb{F}_{j \in i^+, \varphi \in \Omega_j}\left[V_j^k(x^\prime_k, \varphi)\right] + \frac{d}{dx^\prime}\mathbb{F}_{j \in i^+, \varphi \in \Omega_j}\left[V_j^k(x^\prime_k, \varphi)\right]^\top (x^\prime - x^\prime_k),\quad k=1,\ldots,K\]

as:

\[\theta \ge \mathbb{E}_{q_k}\left[V_j^k(x^\prime_k, \varphi) + \frac{d}{dx^\prime}V_j^k(x^\prime_k, \varphi)^\top (x^\prime - x^\prime_k)\right] - \alpha(p, q_k),\quad k=1,\ldots,K,\]

where $q_k = \mathrm{arg}\sup\limits_{q \in\mathcal{M}(p)} \mathbb{E}_q[V_j^k(x_k^\prime, \varphi)] - \alpha(p, q)$.

Therefore, we can formulate a risk-averse decision rule as:

\[\begin{aligned} V_i^K(x, \omega) = \min\limits_{\bar{x}, x^\prime, u} \;\; & C_i(\bar{x}, u, \omega) + \theta\\ & x^\prime = T_i(\bar{x}, u, \omega) \\ & u \in U_i(\bar{x}, \omega) \\ & \bar{x} = x \\ & \theta \ge \mathbb{E}_{q_k}\left[V_j^k(x^\prime_k, \varphi) + \frac{d}{dx^\prime}V_j^k(x^\prime_k, \varphi)^\top (x^\prime - x^\prime_k)\right] - \alpha(p, q_k),\quad k=1,\ldots,K \\ & \theta \ge M. \end{aligned}\]

where $q_k = \mathrm{arg}\sup\limits_{q \in\mathcal{M}(p)} \mathbb{E}_q[V_j^k(x_k^\prime, \varphi)] - \alpha(p, q)$.

Thus, to implement risk-averse SDDP, all we need to do is modify the backward pass to include this calculation of $q_k$, form the cut using $q_k$ instead of $p$, and subtract the penalty term $\alpha(p, q_k)$.

Implementation

Now we're ready to implement our risk-averse version of SDDP.

As a prerequisite, we need most of the code from Introductory theory.

Click to view code from the tutorial "Introductory theory".
struct State
    in::JuMP.VariableRef
    out::JuMP.VariableRef
end

struct Uncertainty
    parameterize::Function
    Ω::Vector{Any}
    P::Vector{Float64}
end

struct Node
    subproblem::JuMP.Model
    states::Dict{Symbol,State}
    uncertainty::Uncertainty
    cost_to_go::JuMP.VariableRef
end

struct PolicyGraph
    nodes::Vector{Node}
    arcs::Vector{Dict{Int,Float64}}
end

function Base.show(io::IO, model::PolicyGraph)
    println(io, "A policy graph with $(length(model.nodes)) nodes")
    println(io, "Arcs:")
    for (from, arcs) in enumerate(model.arcs)
        for (to, probability) in arcs
            println(io, "  $(from) => $(to) w.p. $(probability)")
        end
    end
    return
end

function PolicyGraph(
    subproblem_builder::Function;
    graph::Vector{Dict{Int,Float64}},
    lower_bound::Float64,
    optimizer,
)
    nodes = Node[]
    for t in 1:length(graph)
        model = JuMP.Model(optimizer)
        states, uncertainty = subproblem_builder(model, t)
        JuMP.@variable(model, cost_to_go >= lower_bound)
        obj = JuMP.objective_function(model)
        JuMP.@objective(model, Min, obj + cost_to_go)
        if length(graph[t]) == 0
            JuMP.fix(cost_to_go, 0.0; force = true)
        end
        push!(nodes, Node(model, states, uncertainty, cost_to_go))
    end
    return PolicyGraph(nodes, graph)
end

function sample_uncertainty(uncertainty::Uncertainty)
    r = rand()
    for (p, ω) in zip(uncertainty.P, uncertainty.Ω)
        r -= p
        if r < 0.0
            return ω
        end
    end
    return error("We should never get here because P should sum to 1.0.")
end

function sample_next_node(model::PolicyGraph, current::Int)
    if length(model.arcs[current]) == 0
        return nothing
    else
        r = rand()
        for (to, probability) in model.arcs[current]
            r -= probability
            if r < 0.0
                return to
            end
        end
        return nothing
    end
end

function forward_pass(model::PolicyGraph, io::IO = stdout)
    incoming_state =
        Dict(k => JuMP.fix_value(v.in) for (k, v) in model.nodes[1].states)
    simulation_cost = 0.0
    trajectory = Tuple{Int,Dict{Symbol,Float64}}[]
    t = 1
    while t !== nothing
        node = model.nodes[t]
        ω = sample_uncertainty(node.uncertainty)
        node.uncertainty.parameterize(ω)
        for (k, v) in incoming_state
            JuMP.fix(node.states[k].in, v; force = true)
        end
        JuMP.optimize!(node.subproblem)
        if JuMP.termination_status(node.subproblem) != JuMP.MOI.OPTIMAL
            error("Something went terribly wrong!")
        end
        outgoing_state = Dict(k => JuMP.value(v.out) for (k, v) in node.states)
        stage_cost =
            JuMP.objective_value(node.subproblem) - JuMP.value(node.cost_to_go)
        simulation_cost += stage_cost
        incoming_state = outgoing_state
        push!(trajectory, (t, outgoing_state))
        t = sample_next_node(model, t)
    end
    return trajectory, simulation_cost
end

function upper_bound(model::PolicyGraph; replications::Int)
    simulations = [forward_pass(model, devnull) for i in 1:replications]
    z = [s[2] for s in simulations]
    μ = Statistics.mean(z)
    tσ = 1.96 * Statistics.std(z) / sqrt(replications)
    return μ, tσ
end

function lower_bound(model::PolicyGraph)
    node = model.nodes[1]
    bound = 0.0
    for (p, ω) in zip(node.uncertainty.P, node.uncertainty.Ω)
        node.uncertainty.parameterize(ω)
        JuMP.optimize!(node.subproblem)
        bound += p * JuMP.objective_value(node.subproblem)
    end
    return bound
end

function evaluate_policy(
    model::PolicyGraph;
    node::Int,
    incoming_state::Dict{Symbol,Float64},
    random_variable,
)
    the_node = model.nodes[node]
    the_node.uncertainty.parameterize(random_variable)
    for (k, v) in incoming_state
        JuMP.fix(the_node.states[k].in, v; force = true)
    end
    JuMP.optimize!(the_node.subproblem)
    return Dict(
        k => JuMP.value.(v) for
        (k, v) in JuMP.object_dictionary(the_node.subproblem)
    )
end
evaluate_policy (generic function with 1 method)

First, we need to modify the backward pass to compute the cuts using the risk-averse subgradient theorem:

function backward_pass(
    model::PolicyGraph,
    trajectory::Vector{Tuple{Int,Dict{Symbol,Float64}}},
    io::IO = stdout;
    risk_measure::AbstractRiskMeasure,
)
    println(io, "| Backward pass")
    for i in reverse(1:length(trajectory))
        index, outgoing_states = trajectory[i]
        node = model.nodes[index]
        println(io, "| | Visiting node $(index)")
        if length(model.arcs[index]) == 0
            continue
        end
        # =====================================================================
        # New! Create vectors to store the cut expressions, V(x,ω) and p:
        cut_expressions, V_ω, p = JuMP.AffExpr[], Float64[], Float64[]
        # =====================================================================
        for (j, P_ij) in model.arcs[index]
            next_node = model.nodes[j]
            for (k, v) in outgoing_states
                JuMP.fix(next_node.states[k].in, v; force = true)
            end
            for (pφ, φ) in zip(next_node.uncertainty.P, next_node.uncertainty.Ω)
                next_node.uncertainty.parameterize(φ)
                JuMP.optimize!(next_node.subproblem)
                V = JuMP.objective_value(next_node.subproblem)
                dVdx = Dict(
                    k => JuMP.reduced_cost(v.in) for (k, v) in next_node.states
                )
                # =============================================================
                # New! Construct and append the expression
                # `V_j^K(x_k, φ) + dVdx_j^K(x'_k, φ)ᵀ(x - x_k)` to the list of
                # cut expressions.
                push!(
                    cut_expressions,
                    JuMP.@expression(
                        node.subproblem,
                        V + sum(
                            dVdx[k] * (x.out - outgoing_states[k]) for
                            (k, x) in node.states
                        ),
                    )
                )
                # Add the objective value to Z:
                push!(V_ω, V)
                # Add the probability to p:
                push!(p, P_ij * pφ)
                # =============================================================
            end
        end
        # =====================================================================
        # New! Using the solutions in V_ω, compute q and α:
        q, α = dual_risk_inner(risk_measure, V_ω, p)
        println(io, "| | | Z = ", Z)
        println(io, "| | | p = ", p)
        println(io, "| | | q = ", q)
        println(io, "| | | α = ", α)
        # Then add the cut:
        c = JuMP.@constraint(
            node.subproblem,
            node.cost_to_go >=
            sum(q[i] * cut_expressions[i] for i in 1:length(q)) - α
        )
        # =====================================================================
        println(io, "| | | Adding cut : ", c)
    end
    return nothing
end
backward_pass (generic function with 2 methods)

We also need to update the train loop of SDDP to pass a risk measure to the backward pass:

function train(
    model::PolicyGraph;
    iteration_limit::Int,
    replications::Int,
    # =========================================================================
    # New! Add a risk_measure argument
    risk_measure::AbstractRiskMeasure,
    # =========================================================================
    io::IO = stdout,
)
    for i in 1:iteration_limit
        println(io, "Starting iteration $(i)")
        outgoing_states, _ = forward_pass(model, io)
        # =====================================================================
        # New! Pass the risk measure to the backward pass.
        backward_pass(model, outgoing_states, io; risk_measure = risk_measure)
        # =====================================================================
        println(io, "| Finished iteration")
        println(io, "| | lower_bound = ", lower_bound(model))
    end
    μ, tσ = upper_bound(model; replications = replications)
    println(io, "Upper bound = $(μ) ± $(tσ)")
    return
end
train (generic function with 1 method)

Risk-averse bounds

Warning

This section is important.

When we had a risk-neutral policy (i.e., we only used the expectation risk measure), we discussed how we could form valid lower and upper bounds.

The upper bound is still valid as a Monte Carlo simulation of the expected cost of the policy. (Although this upper bound doesn't capture the change in the policy we wanted to achieve, namely that the impact of the worst outcomes were reduced.)

However, if we use a different risk measure, the lower bound is no longer valid!

We can still calculate a "lower bound" as the objective of the first-stage approximated subproblem, and this will converge to a finite value. However, we can't meaningfully interpret it as a bound with respect to the optimal policy. Therefore, it's best to just ignore the lower bound when training a risk-averse policy.

Example: risk-averse hydro-thermal scheduling

Now it's time for an example. We create the same problem as Introductory theory:

model = PolicyGraph(
    graph = [Dict(2 => 1.0), Dict(3 => 1.0), Dict{Int,Float64}()],
    lower_bound = 0.0,
    optimizer = HiGHS.Optimizer,
) do subproblem, t
    JuMP.set_silent(subproblem)
    JuMP.@variable(subproblem, volume_in == 200)
    JuMP.@variable(subproblem, 0 <= volume_out <= 200)
    states = Dict(:volume => State(volume_in, volume_out))
    JuMP.@variables(subproblem, begin
        thermal_generation >= 0
        hydro_generation >= 0
        hydro_spill >= 0
        inflow
    end)
    JuMP.@constraints(
        subproblem,
        begin
            volume_out == volume_in + inflow - hydro_generation - hydro_spill
            demand_constraint, thermal_generation + hydro_generation == 150.0
        end
    )
    fuel_cost = [50.0, 100.0, 150.0]
    JuMP.@objective(subproblem, Min, fuel_cost[t] * thermal_generation)
    uncertainty =
        Uncertainty([0.0, 50.0, 100.0], [1 / 3, 1 / 3, 1 / 3]) do ω
            return JuMP.fix(inflow, ω)
        end
    return states, uncertainty
end
A policy graph with 3 nodes
Arcs:
  1 => 2 w.p. 1.0
  2 => 3 w.p. 1.0

Then we train a risk-averse policy, passing a risk measure to train:

train(
    model;
    iteration_limit = 3,
    replications = 100,
    risk_measure = Entropic(1.0),
)
Starting iteration 1
| Backward pass
| | Visiting node 3
| | Visiting node 2
| | | Z = [1.0, 2.0, 3.0, 4.0]
| | | p = [0.3333333333333333, 0.3333333333333333, 0.3333333333333333]
| | | q = [1.0, 0.0, 0.0]
| | | α = 1.0986122886681098
| | | Adding cut : 150 volume_out + cost_to_go ≥ 22498.901387711332
| | Visiting node 1
| | | Z = [1.0, 2.0, 3.0, 4.0]
| | | p = [0.3333333333333333, 0.3333333333333333, 0.3333333333333333]
| | | q = [1.0, 0.0, 0.0]
| | | α = 1.0986122886681098
| | | Adding cut : 150 volume_out + cost_to_go ≥ 37497.802775422664
| Finished iteration
| | lower_bound = 12497.802775422664
Starting iteration 2
| Backward pass
| | Visiting node 3
| | Visiting node 2
| | | Z = [1.0, 2.0, 3.0, 4.0]
| | | p = [0.3333333333333333, 0.3333333333333333, 0.3333333333333333]
| | | q = [0.5999999999999176, 0.2000000000000412, 0.2000000000000412]
| | | α = 0.14834174943478465
| | | Adding cut : 89.99999999998764 volume_out + cost_to_go ≥ 13499.851658248712
| | Visiting node 1
| | | Z = [1.0, 2.0, 3.0, 4.0]
| | | p = [0.3333333333333333, 0.3333333333333333, 0.3333333333333333]
| | | q = [1.0, 0.0, 0.0]
| | | α = 1.0986122886681098
| | | Adding cut : 100 volume_out + cost_to_go ≥ 29998.594667538695
| Finished iteration
| | lower_bound = 14998.594667538693
Starting iteration 3
| Backward pass
| | Visiting node 3
| | Visiting node 2
| | | Z = [1.0, 2.0, 3.0, 4.0]
| | | p = [0.3333333333333333, 0.3333333333333333, 0.3333333333333333]
| | | q = [0.3903334174068186, 0.3048332912965907, 0.3048332912965907]
| | | α = 0.007126620949088093
| | | Adding cut : 58.55001261102279 volume_out + cost_to_go ≥ 8782.49476503247
| | Visiting node 1
| | | Z = [1.0, 2.0, 3.0, 4.0]
| | | p = [0.3333333333333333, 0.3333333333333333, 0.3333333333333333]
| | | q = [1.0, 0.0, 0.0]
| | | α = 1.0986122886681098
| | | Adding cut : 100 volume_out + cost_to_go ≥ 29998.59466753869
| Finished iteration
| | lower_bound = 14998.594667538693
Upper bound = 9999.44677098547 ± 897.2618923237071

Finally, evaluate the decision rule:

evaluate_policy(
    model;
    node = 1,
    incoming_state = Dict(:volume => 150.0),
    random_variable = 75,
)
Dict{Symbol, Float64} with 8 entries:
  :volume_out         => 200.0
  :demand_constraint  => 150.0
  :hydro_spill        => 0.0
  :inflow             => 75.0
  :volume_in          => 150.0
  :thermal_generation => 125.0
  :hydro_generation   => 25.0
  :cost_to_go         => 9998.59
Info

For this trivial example, the risk-averse policy isn't very different from the policy obtained using the expectation risk-measure. If you try it on some bigger/more interesting problems, you should see the expected cost increase, and the upper tail of the policy decrease.