Concept Architecture
Concept
Theoretically, Numerical Optimisation is a mathematical framework for identifying the values of decision variables or model parameters that maximise or minimise an objective function when an analytical solution is unavailable or impractical. It is founded on optimisation theory and numerical analysis, where iterative computational algorithms are used to search the solution space while satisfying any specified constraints. The concept exists because many health economic models involve complex nonlinear functions, stochastic processes, or high-dimensional parameter spaces that cannot be solved algebraically.
Mathematically, numerical optimisation formulates the problem as the optimisation of an objective function over a feasible region defined by equality and inequality constraints. Depending on the characteristics of the objective function, recognised optimisation algorithms may use derivatives, direct search techniques, or stochastic search methods to iteratively improve the solution until predefined convergence criteria are met. The resulting solution represents a local or global optimum according to the optimisation method and problem structure.
In practice, numerical optimisation is implemented by defining the objective function, specifying model parameters and constraints, selecting an appropriate optimisation algorithm, and applying convergence criteria. In health economics it is widely used for model calibration, maximum likelihood estimation, parameter estimation, resource allocation, budget optimisation, treatment pathway optimisation, and fitting survival or disease progression models to observed data.
Purpose
Used to determine optimal parameter values or resource allocation decisions by iteratively maximising or minimising objective functions that cannot be solved analytically.
Mathematical Formulae
Primary Formula
min? f(x)
subject to
g?(x) � 0, i = 1, ?, m
h?(x) = 0, j = 1, ?, p
where:
- f(x) = objective function
- x = decision variable vector
- g?(x) = inequality constraints
- h?(x) = equality constraints
Supporting Formulae
First-order optimality condition:
?f(x) = 0*
Gradient descent update:
x??? = x? ? �?f(x?)
where:
- ?f(x) = gradient of the objective function
- � = step size
- x* = optimal solution
Related Mathematical Methods
- Gradient descent
- Newton-Raphson method
- Nelder-Mead method
- Linear programming
- Nonlinear programming
- Integer programming
- Sequential quadratic programming
- Simulated annealing
- Genetic algorithms
Example
A health economist calibrates a Markov model by minimising the sum of squared differences between observed and predicted survival probabilities.
The objective function is
min ?????? (S????????? ? S??????)�.
An optimisation algorithm adjusts the transition probabilities iteratively. The objective function decreases from 12.84 at the initial parameter values to 0.41 after convergence, indicating substantially improved agreement between model predictions and observed survival data.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUMXMY2 | =SUMXMY2(B2:B11,C2:C11) | Calculates the objective function as the sum of squared errors during model calibration. |
| SUMPRODUCT | =SUMPRODUCT(B2:B11,C2:C11) | Computes weighted objective functions and optimisation criteria. |
| Solver Add-in | Minimise objective cell by changing parameter cells | Performs numerical optimisation of model parameters or resource allocation decisions. |
| ABS | =ABS(B2-C2) | Monitors prediction errors during optimisation. |
VBA (Optional)
A VBA procedure can automate repeated optimisation runs across alternative starting values, convergence tolerances, or model scenarios to improve calibration efficiency.
Sources
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
- Nocedal J, Wright SJ. Numerical Optimization. Springer.
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes.
- NICE. Health Technology Evaluation Manual.
- ISPOR Good Practice Reports on model calibration and decision-analytic modelling.
Related Concepts (6)
Library
Publications
1
A Taxonomy of Model Structures for Economic Evaluation of Health Technologies — Brennan, Chick & Davies, Vol. 15, No. 12 ed., 2006 (Health Economics)
An influential paper classifying decision-analytic model structures along axes of expected value vs randomness, entity heterogeneity, and Markovian vs non-Markovian structure — providing a framework for choosing between decision trees, Markov cohort models, microsimulation, discrete event simulation and system dynamics.
Journal ArticleView source →
Frequently Asked Questions (6)
What is numerical optimisation?
Computational techniques for finding the input values that minimise or maximise an objective function when no analytical solution is available.
Source: Nocedal & Wright 2006
Why can numerical optimisation not always be avoided by algebra?
For simple problems the best values can be found by setting a derivative to zero and solving, but many objectives arising from models are too complex, discontinuous, or lacking a closed form for that to work. Numerical optimisation instead searches the space of possible inputs by trial and refinement, evaluating the objective repeatedly and moving toward better values. It is the practical route to an answer when the mathematics of the problem admits no direct solution. Nocedal and Wright (2006) set out these methods.
Source: Nocedal & Wright 2006
Why is numerical optimisation needed?
Numerical optimisation is needed because many optimisation problems cannot be solved analytically: the objective may be complex, non-linear, or defined only by a computation such as a simulation, so the optimum cannot be found in closed form. In such cases, iterative numerical methods search for the best input values. This arises constantly in statistics, where models are fitted by optimising a criterion, and in modelling and planning, where the best parameters or decisions must be found when no formula gives them directly.
Source: Nocedal & Wright 2006
What are the main types of numerical optimisation method?
The main types include gradient-based methods, which use the objective's derivatives to move in the direction that most improves it, such as gradient descent and Newton-type methods; and derivative-free methods, which use only function values, such as the Nelder-Mead simplex, grid and random search, and population-based methods. Some methods seek a local optimum efficiently, while others, such as simulated annealing and genetic algorithms, aim to find a global optimum in difficult landscapes. The choice depends on the problem.
Source: Nocedal & Wright 2006
What is the difference between local and global optimisation?
Local optimisation finds an optimum in the neighbourhood of the starting point, which may be a local rather than the global best if the objective has several optima, while global optimisation seeks the overall best across the whole space. Gradient-based methods typically find local optima efficiently but can be trapped by them, whereas global methods, such as simulated annealing or genetic algorithms, explore more broadly to escape local optima, at greater computational cost. The distinction matters where the objective is not simple.
Source: Nocedal & Wright 2006
Where is numerical optimisation used in health economics?
In health economics, numerical optimisation is used to fit and calibrate models, for instance finding parameter values that make a model match observed data or targets, to estimate statistical models by optimising a likelihood or fit criterion, and to solve resource allocation problems that maximise benefit subject to constraints. Wherever the best values of parameters or decisions must be found and no formula gives them, numerical optimisation provides the computational means, making it a basic tool in modelling and estimation.
Source: Nocedal & Wright 2006
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 2 Oct 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-EM-DM-072
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