01 · Lesson 12
Optimisation
Optimisation searches for an admissible choice that minimises or maximises a stated objective. In mathematical biology it can estimate parameters, design treatments or allocate limited resources—but the answer is only as meaningful as the objective, constraints and model.
Scenario: fitting a growth model
Population measurements are available at times \(t_1,\ldots,t_n\). A logistic model predicts \(N(t;\theta)\) using parameters \(\theta=(r,K)\). We seek parameter values whose predictions are close to the observations.
A least-squares objective is
The estimate is an admissible minimiser \(\widehat\theta\) of \(J\).
Four essential components
| Component | Question |
|---|---|
| Decision variables | What may be changed? |
| Objective function | What numerical quantity is minimised or maximised? |
| Constraints | Which choices are permitted? |
| Model and data | How does a choice produce a predicted consequence? |
For parameter fitting, constraints such as \(r>0\) and \(K>0\) preserve biological meaning.
Local and global optima
A local minimum is better than sufficiently nearby choices. A global minimum is at least as good as every admissible choice. Nonlinear biological objectives may contain several local minima, flat regions or boundaries.
Trying different starting values and inspecting the objective surface can reveal sensitivity, but does not always prove that the global optimum was found.
Derivative conditions
For a differentiable unconstrained objective, an interior candidate optimum satisfies
In one dimension, \(J'(\theta^*)=0\) identifies a stationary point. A positive second derivative supports a local minimum; a negative one supports a local maximum. Boundary optima need not have zero derivative.
Parameter estimation is not parameter identification
An optimiser always returning a number does not mean the data determine that parameter reliably. Several parameter combinations may produce nearly identical trajectories. This is an identifiability problem.
Report the fitted value together with uncertainty, sensitivity to starting values, residual behaviour and the assumptions connecting observations to the model.
Optimising an intervention
A vaccination strategy might minimise
where \(u\) describes the strategy, the integral measures disease burden and \(C(u)\) measures intervention cost. Weights \(w_1,w_2\) encode a value judgement about competing outcomes; mathematics does not choose those values neutrally.
Constraints may represent vaccine supply, treatment capacity, timing, safety or equity. Omitting an important constraint can make a mathematically optimal answer biologically or ethically unusable.
Uncertainty and robustness
An intervention optimised for one estimated parameter set may perform poorly when parameters differ. Robust or stochastic optimisation evaluates choices across uncertainty rather than only at a single best estimate. Scenario and sensitivity analyses should accompany the reported optimum.
Conditions and limitations
- Define every decision variable, objective term, unit and constraint.
- Distinguish fitting data from predicting new data.
- Check multiple initial guesses for nonlinear problems.
- Investigate identifiability and numerical convergence.
- Propagate parameter and observation uncertainty.
- Do not describe an objective-dependent solution as universally “best.”
What this lesson adds
You can now formulate an optimisation problem, distinguish local and global solutions, understand derivative conditions and constraints, and interpret fitted parameters or interventions with appropriate uncertainty and biological caution.
Continue along the learning path
Use these mathematical tools to learn how biological questions become model assumptions, states, rates and equations.