Concept Architecture
Concept
Theoretically, Newton Optimisation is a second-order numerical optimisation method used to locate stationary points of a differentiable objective function by exploiting both first- and second-order derivative information. It is founded on Newton's method for solving nonlinear equations and extends this principle to optimisation by using the gradient and Hessian matrix to determine parameter updates. The method exists to achieve rapid convergence, particularly when the objective function is locally quadratic and twice continuously differentiable.
Mathematically, Newton Optimisation iteratively updates parameter estimates by moving in a direction determined by the inverse Hessian matrix multiplied by the gradient vector. Unlike first-order methods such as Gradient Descent, Newton Optimisation accounts for the local curvature of the objective function, often resulting in quadratic convergence near the optimum. The algorithm continues until the gradient approaches zero or another convergence criterion is satisfied.
In practice, Newton Optimisation is implemented in statistical estimation, maximum likelihood estimation, nonlinear regression and calibration of health economic models. It is widely used where objective functions are smooth and second derivatives can be computed or approximated efficiently. In health economics, the method underpins parameter estimation in econometric models, calibration of simulation models and optimisation problems requiring accurate and computationally efficient convergence.
Purpose
Used to efficiently minimise or maximise differentiable objective functions by incorporating both gradient and curvature information, supporting parameter estimation, model calibration and numerical optimisation in health economic analysis.
Mathematical Formulae
Primary Formula
???? = ?? ? H(??)???J(??)
Supporting Formulae
Gradient condition at optimum:
?J(??) = 0
Hessian matrix:
H(?) = ?�J(?) / ?????
Second-order Taylor approximation:
J(? + ??) � J(?) + ?J(?)??? + (1/2)???H(?)??
Related Mathematical Methods
- Newton-Raphson Method
- Gradient Descent
- Quasi-Newton Methods
- BFGS Algorithm
- Conjugate Gradient Method
- Maximum Likelihood Estimation
- Numerical Optimisation
Example
A health economist estimates the parameters of a nonlinear cost prediction model by minimising the negative log-likelihood.
Current parameter estimate:
?? = 2.50
Gradient:
?J(??) = 4.0
Hessian:
H(??) = 20.0
Parameter update:
?? = 2.50 ? (1/20.0) ? 4.0
?? = 2.30
The optimisation continues until the gradient is sufficiently close to zero, indicating convergence to the optimum.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| Solver Add-in | Minimise objective cell by changing parameter cells | Performs Newton-based optimisation where supported by the optimisation engine. |
| SUMPRODUCT | =SUMPRODUCT(XRange,BetaRange) | Evaluates the objective function. |
| MMULT | =MMULT(Matrix1,Matrix2) | Performs matrix multiplication for Hessian calculations. |
| MINVERSE | =MINVERSE(HessianRange) | Computes the inverse Hessian matrix. |
| ABS | =ABS(NewEstimate-OldEstimate) | Assesses convergence between successive iterations. |
VBA (Optional)
VBA can automate Newton optimisation by iteratively evaluating gradients and Hessian matrices, updating parameters and terminating when convergence criteria are satisfied.
Sources
- Nocedal J, Wright SJ. Numerical Optimization. Springer.
- Boyd S, Vandenberghe L. Convex Optimization. Cambridge University Press.
- Dennis JE, Schnabel RB. Numerical Methods for Unconstrained Optimization and Nonlinear Equations.
- 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.
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes: The Art of Scientific Computing.
Related Concepts (5)
Library
Publications
1
Numerical Analysis of Parameterized Nonlinear Equations — Werner C. Rheinboldt, 1st Edition ed., 1986 (John Wiley & Sons)
A specialist numerical-analysis reference on iterative solution, continuation and computation for parameterized nonlinear systems, linked here to the ontology's numerical-method and optimization branch.
BookView source →
Frequently Asked Questions (6)
What is Newton optimisation?
An iterative optimisation method that uses first and second derivatives to estimate the location of a function's minimum or maximum.
Source: Nocedal J, Wright SJ. Numerical Optimization. 2nd ed. Springer; 2006. doi:10.1007/978-0-387-40065-5.
How does Newton optimisation locate a minimum or maximum?
Newton optimisation locates a minimum or maximum by using first and second derivatives to estimate where the function turns, then iterating towards that point. The first derivative indicates the direction of change and the second describes its curvature, and together they guide each step. This use of first and second derivatives to iterate towards an extremum is how Newton optimisation proceeds By drawing on curvature as well as slope, the method can home in on an extremum quickly near it, at the cost of needing second-derivative information that simpler methods do without.
Source: Nocedal J, Wright SJ. Numerical Optimization. 2nd ed. Springer; 2006. doi:10.1007/978-0-387-40065-5.
What derivatives does Newton optimisation use?
Newton optimisation uses first and second derivatives, drawing on the slope and the curvature of the function to estimate the location of its minimum or maximum. The first derivative gives the direction and the second the rate at which that direction changes. Relying on both first and second derivatives is what distinguishes Newton optimisation By drawing on curvature as well as slope, the method can home in on an extremum quickly near it, at the cost of needing second-derivative information that simpler methods do without.
Source: Nocedal J, Wright SJ. Numerical Optimization. 2nd ed. Springer; 2006. doi:10.1007/978-0-387-40065-5.
Why does Newton optimisation use the second derivative?
Newton optimisation uses the second derivative because it describes the curvature of the function, which lets each step better estimate where the minimum or maximum lies than the slope alone would. Combining this curvature information with the first derivative sharpens the iteration. This reliance on the second derivative for curvature is a defining feature of Newton optimisation By drawing on curvature as well as slope, the method can home in on an extremum quickly near it, at the cost of needing second-derivative information that simpler methods do without.
Source: Nocedal J, Wright SJ. Numerical Optimization. 2nd ed. Springer; 2006. doi:10.1007/978-0-387-40065-5.
What kind of method is Newton optimisation?
Newton optimisation is an iterative optimisation method, refining an estimate step by step using first and second derivatives to approach a function's minimum or maximum. Each iteration improves on the last using the derivative information. Being an iterative method guided by first and second derivatives is what Newton optimisation is By drawing on curvature as well as slope, the method can home in on an extremum quickly near it, at the cost of needing second-derivative information that simpler methods do without.
Source: Nocedal J, Wright SJ. Numerical Optimization. 2nd ed. Springer; 2006. doi:10.1007/978-0-387-40065-5.
How does Newton optimisation relate to gradient descent?
Newton optimisation uses first and second derivatives to estimate the location of a function's minimum or maximum, while gradient descent moves towards a minimum by adjusting parameters in the direction opposite to the gradient, using the first derivative only. Newton optimisation adds curvature information that gradient descent does not use. The two are connected as iterative optimisation methods differing in how much derivative information they draw on By drawing on curvature as well as slope, the method can home in on an extremum quickly near it, at the cost of needing second-derivative information that simpler methods do without.
Source: Nocedal J, Wright SJ. Numerical Optimization. 2nd ed. Springer; 2006. doi:10.1007/978-0-387-40065-5.
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 26 May 2026
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/newton-optimisation
- Term code
- CS-OP-UO-002
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