Topic 26 – Stochastic Optimisation

1. Deterministic vs stochastic optimisation

Deterministic:

Single “world”: choose x to minimise f(x) with constraints.

Stochastic:

Many possible futures. You choose one decision x now, but performance depends on a random outcome ω.

We represent ω via scenarios:

• Scenarios s = 1,…,S with probabilities pₛ,
• Scenario loss (or cost) Lₛ(x).

Basic stochastic program:

$$\min_x \; \mathbb{E}[L(x)] = \sum_s p_s L_s(x) \quad \text{s.t.} \quad x \in \mathcal{X}$$

Examples:

• Choose forward hedges to minimise expected portfolio loss under many price / load scenarios.
• Choose a power procurement mix under many demand / spot price paths.

2. CVaR as an objective (tail-focused)

CVaRₐ (Expected Shortfall at level α, e.g. 97.5%):

• Focuses on average of the worst (1−α)% losses.
• Better tail risk measure than VaR.

Rockafellar–Uryasev trick (scenario form):

Introduce η (loss “threshold”) and auxiliary zₛ:

$$\min_{x, \eta, z} \; \eta + \frac{1}{(1-\alpha) S} \sum_s z_s$$

subject to

$$z_s \ge 0, \quad z_s \ge L_s(x) - \eta \quad \forall s.$$

Interpretation:

• η ≈ VaR (boundary of bad losses),
• zₛ captures excess loss above η,
• Objective = “threshold + average excess in the tail”.

If Lₛ(x) is linear in x, this is a convex program → numerically nice.

Use cases:

• Portfolio allocation with CVaR objective instead of variance.
• Hedging decisions for power/gas portfolios under price + volume scenarios.

3. Chance constraints (probabilistic constraints)

Example constraint:

“Probability that loss exceeds $L^*$ must be $\le 5\%$.”

Mathematically:

$$\mathbb{P}(L(x) > L^*) \le \varepsilon.$$

Scenario approximation:

$$\frac{1}{S} \sum_s \mathbf{1}\{L_s(x) > L^*\} \le \varepsilon.$$

This is non-smooth (indicator). In practice:

• Often replaced by CVaR constraints, which are conservative:
  • If $\operatorname{CVaR}_{\alpha}(L(x)) \le L^*$, then big tail events are controlled.
• Or implemented approximately with penalisation of violations.

Risk-language interpretation:

“Instead of just limiting expected loss or variance, we directly encode: ‘Bad events above $L^*$ must be rare’.”

4. Power / commodities flavour

Typical stochastic optimisation objects:

• Hedging a load portfolio:
  • Scenarios over:
    • Spot prices,
    • Forward curves,
    • Demand (volume).
  • Decision x = vector of forwards, options, structured deals.
  • Objective: minimise CVaR of total cost over winter.
• Hydro / storage management:
  • Scenarios over inflows + prices.
  • DP / stochastic programming blends with CVaR or chance constraints:
    • Avoid reservoir spill (wasted inflow),
    • Avoid shortage with high probability.

Key phrases to drop:

• “Scenario-based CVaR optimisation for winter hedging.”
• “Chance constraints to ensure supply shortfalls are rare.”

5. Interview soundbite

“Stochastic optimisation treats decisions under many future scenarios. We build scenario P&L or cost Lₛ(x) and then choose x to minimise expected loss or a tail measure like CVaR. CVaR is attractive because it has a convex reformulation: we introduce a loss threshold η and auxiliary variables for tail losses, which makes CVaR-based portfolio or hedging optimisation tractable. For power and gas, we use price and load scenarios and often care about constraints like ‘cost above X should happen with probability ≤ 5%’, which we implement via chance constraints or CVaR limits.”