Concept Architecture
Concept
Theoretically, Nonlinear Programming is a mathematical optimisation framework used to maximise or minimise an objective function when the objective, one or more constraints, or both are nonlinear. It is founded on optimisation theory, differential calculus and convex analysis and extends linear programming to problems involving curvature, interactions, diminishing returns and other non-proportional relationships. Nonlinear Programming is central to operations research, economics and computational decision analysis.
Mathematically, a Nonlinear Programming problem is expressed in terms of decision variables, a nonlinear objective function and equality or inequality constraints. Candidate optima are commonly characterised using first-order and second-order conditions together with the Karush?Kuhn?Tucker conditions. Convex problems possess globally optimal solutions under appropriate regularity conditions, whereas non-convex problems may contain multiple local optima and require specialised numerical algorithms.
In practice, Nonlinear Programming is used in health economics to optimise treatment allocation, calibrate decision models, estimate constrained econometric models and allocate healthcare resources where costs, outcomes or constraints are nonlinear. It supports complex analyses involving diminishing marginal returns, nonlinear production functions, risk constraints and interacting policy variables.
Purpose
Used to optimise nonlinear objective functions subject to constraints, represent complex healthcare relationships, support model calibration and improve resource allocation in health economic analysis.
Mathematical Formulae
Primary Formula
Minimise or maximise
f(x)
Subject to
g?(x) � 0,?i = 1, ?, m
h?(x) = 0,?j = 1, ?, p
where:
- x = decision variable vector
- f(x) = nonlinear objective function
- g?(x) = inequality constraints
- h?(x) = equality constraints
Supporting Formulae
Lagrangian Function
L(x, ?, ?) = f(x) + �???? ??g?(x) + �???? ??h?(x)
Karush?Kuhn?Tucker Conditions
??L(x, ?, ?) = 0
g?(x) � 0
?? � 0
??g?(x) = 0
h?(x) = 0
Related Mathematical Methods
- Constrained optimisation
- Karush?Kuhn?Tucker conditions
- Interior-Point Method
- Sequential quadratic programming
- Gradient descent
- Convex optimisation
Example
A healthcare authority allocates resources between two programmes to maximise health benefit:
Maximise
Z = 10�x? + 14�x?
Subject to
4x? + 6x? � 240
x? � 0
x? � 0
The square-root terms represent diminishing marginal health returns from additional investment. Because the objective function is nonlinear, the allocation problem requires Nonlinear Programming rather than Linear Programming.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUMPRODUCT | =SUMPRODUCT(Costs,DecisionVariables) | Calculate resource use under nonlinear optimisation constraints. |
| SQRT | =10*SQRT(B2)+14*SQRT(C2) | Represent diminishing marginal health returns. |
| Solver | Select GRG Nonlinear and optimise the objective | Solve constrained nonlinear healthcare allocation problems. |
| IF | =IF(D2<=E2,"Feasible","Constraint Violated") | Check whether nonlinear solutions satisfy constraints. |
| ABS | =ABS(F2-G2) | Monitor convergence between successive optimisation iterations. |
VBA (Optional)
Automate nonlinear optimisation through Excel Solver, evaluate alternative starting values, monitor convergence and generate optimal healthcare resource allocation and model calibration results.
Sources
- Nocedal J, Wright SJ. Numerical Optimization.
- Bertsekas DP. Nonlinear Programming.
- Bazaraa MS, Sherali HD, Shetty CM. Nonlinear Programming: Theory and Algorithms.
- Boyd S, Vandenberghe L. Convex Optimization.
- Hillier FS, Lieberman GJ. Introduction to Operations Research.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
Related Concepts (4)
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 nonlinear programming?
A method for optimising an objective function when the objective, constraints or both contain nonlinear relationships.
Source: Nocedal J, Wright SJ. Numerical Optimization. 2nd ed. Springer; 2006. doi:10.1007/978-0-387-40065-5.
What distinguishes nonlinear programming?
Nonlinear programming is distinguished by handling optimisation in which the objective function, the constraints, or both contain nonlinear relationships. Because either part may be nonlinear, it addresses problems that a purely linear method cannot. Coping with nonlinearity in the objective or constraints is what marks a problem as nonlinear programming Because a curved objective or constraint can create several local optima and awkward feasible regions, nonlinear programming is generally harder than the linear case and calls for more elaborate methods.
Source: Nocedal J, Wright SJ. Numerical Optimization. 2nd ed. Springer; 2006. doi:10.1007/978-0-387-40065-5.
Where can the nonlinearity lie in nonlinear programming?
In nonlinear programming, the nonlinearity can lie in the objective function, in the constraints, or in both, since it is a method for optimising when any of these contains nonlinear relationships. It is enough for one part to be nonlinear for the problem to qualify. This possibility of nonlinearity in the objective or constraints defines nonlinear programming Because a curved objective or constraint can create several local optima and awkward feasible regions, nonlinear programming is generally harder than the linear case and calls for more elaborate methods.
Source: Nocedal J, Wright SJ. Numerical Optimization. 2nd ed. Springer; 2006. doi:10.1007/978-0-387-40065-5.
What does nonlinear programming optimise?
Nonlinear programming optimises an objective function, seeking its best value, in the setting where the objective, the constraints, or both involve nonlinear relationships. The optimisation must respect whatever constraints apply, which may themselves be nonlinear. Optimising an objective under possible nonlinearity is what nonlinear programming does Because a curved objective or constraint can create several local optima and awkward feasible regions, nonlinear programming is generally harder than the linear case and calls for more elaborate methods.
Source: Nocedal J, Wright SJ. Numerical Optimization. 2nd ed. Springer; 2006. doi:10.1007/978-0-387-40065-5.
Why can nonlinear programming be harder than linear methods?
Nonlinear programming can be harder because the nonlinear relationships in its objective or constraints allow shapes, such as curved feasible regions and multiple local optima, that linear methods do not encounter. These features complicate the search for a best solution. The nonlinearity that defines nonlinear programming is the source of this added difficulty Because a curved objective or constraint can create several local optima and awkward feasible regions, nonlinear programming is generally harder than the linear case and calls for more elaborate methods.
Source: Nocedal J, Wright SJ. Numerical Optimization. 2nd ed. Springer; 2006. doi:10.1007/978-0-387-40065-5.
How does nonlinear programming relate to linear programming?
Nonlinear programming optimises an objective when the objective, constraints or both contain nonlinear relationships, while linear programming optimises a linear objective subject to linear constraints. Nonlinear programming generalises the linear case to allow nonlinearity. The two are connected as optimisation methods differing in whether their objective and constraints are restricted to linear relationships Because a curved objective or constraint can create several local optima and awkward feasible regions, nonlinear programming is generally harder than the linear case and calls for more elaborate methods.
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
- Term code
- CS-OP-CO-002
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