Topic 25 – Constrained Optimisation

1. Basic picture

Optimisation template:

$$\min_{x} f(x) \quad \text{s.t.} \quad g_i(x) \le 0,\; h_j(x)=0$$

• Decision variables $x$: positions, notional, dispatch, etc.
• Objective $f(x)$: expected P&L, variance, cost, etc.
• Constraints $g_i, h_j$: limits, budgets, regulations, physics.

Risk limits in this language:

• Position limit: $ |x_k| L_k $
• VaR limit (approx): $ (x) V^* $
• Delta bucket: $ |_{}(x)| D^* $

Conceptual point:

“Risk limits define the feasible set; within that set you optimise whatever objective you care about.”

2. Simple quadratic example: mean–variance with constraints

Example (portfolio weights $w$):

$$\min_w \frac{1}{2} w^\top \Sigma w - \lambda \mu^\top w$$ subject to

• Budget: $\sum_i w_i = 1$,
• Box constraints: $w_i^{\text{min}} \le w_i \le w_i^{\text{max}}$.

Here:

• $\Sigma$: covariance matrix (risk),
• $\mu$: expected excess returns.

Risk limits appear as:

• Box constraints (max exposure per asset),
• Linear constraints (sector / factor limits),
• Norm constraints (limit leverage, e.g. $\|w\|_1 \le L$).

3. Lagrange multipliers intuition

Lagrangian for simple equality-constrained problem:

$$\mathcal{L}(x, \lambda) = f(x) + \lambda^\top h(x)$$

• $\lambda$ = Lagrange multipliers (“shadow prices”).
• Interpret $\lambda_j$ as:
  • “Marginal value of relaxing constraint $h_j$ by 1 unit.”

Risk view:

“Tight constraints with large multipliers tell you where risk appetite or regulation is really binding.”

4. Power / commodities flavour

Examples:

• Optimising a hedge book under:
  • Delta / vega / volume limits,
  • Credit and liquidity limits.
• Hydro / storage dispatch under:
  • Min/max generation,
  • Reservoir bounds,
  • Environmental constraints.

You don’t need closed-form KKT in daily work, but you should see:

“Risk limits become formal constraints in an optimisation; good systems treat them as such, not as after-the-fact checks.”

5. Interview soundbite

“Constrained optimisation is just ‘best objective within a feasible region’. For portfolios, we minimise a risk or cost functional subject to position, leverage, and risk limits. In a hydro or structured commodities book, constraints encode capacity, ramp rates, and regulatory or risk limits; Lagrange multipliers tell you which limits are actually binding in the optimum.”