Nonlinear programming

Nonlinear programming
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Nonlinear_programming".

In mathematics, nonlinear programming (NLP) is the process of solving a system of equalities and inequalities, collectively termed constraints, over a set of unknown real variables, along with an objective function to be maximized or minimized, where some of the constraints or the objective function are nonlinear.

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1 Mathematical formulation of the problem
2 Methods for solving the problem
3 Examples
- 3.1 2-dimensional example
- 3.2 3-dimensional example
4 See also
5 References
6 Further reading
7 External links

Mathematical formulation of the problem

The problem can be stated simply as:

$\max_{x \in X}f(x)$ to maximize some variable such as product throughput

$\min_{x \in X}f(x)$ to minimize a cost function

where

$f: R^n \to R$

$X \subseteq R^n.$

Methods for solving the problem

If the objective function f is linear and the constrained space is a polytope, the problem is a linear programming problem, which may be solved using well known linear programming solutions.

If the objective function is concave (maximization problem), or convex (minimization problem) and the constraint set is convex, then the program is called convex and general methods from convex optimization can be used.

Several methods are available for solving nonconvex problems. One approach is to use special formulations of linear programming problems. Another method involves the use of branch and bound techniques, where the program is divided into subclasses to be solved with convex (minimization problem) or linear approximations that form a lower bound on the overall cost within the subdivision. With subsequent divisions, at some point an actual solution will be obtained whose cost is equal to the best lower bound obtained for any of the approximate solutions. This solution is optimal, although possibly not unique. The algorithm may also be stopped early, with the assurance that the best possible solution is within a tolerance from the best point found; such points are called ε-optimal. Terminating to ε-optimal points is typically necessary to ensure finite termination. This is especially useful for large, difficult problems and problems with uncertain costs or values where the uncertainty can be estimated with an appropriate reliability estimation.

Under differentiability and constraint qualifications, the Karush-Kuhn-Tucker (KKT) conditions provide necessary conditions for a solution to be optimal. Under convexity, these conditions are also sufficient.

Examples

2-dimensional example

The intersection of the line with the constrained space represents the solution

A simple problem can be defined by the constraints

x₁ ≥ 0

x₂ ≥ 0

x₁² + x₂² ≥ 1

x₁² + x₂² ≤ 2

with an objective function to be maximized

f(x) = x₁ + x₂

where x = (x₁, x₂).

3-dimensional example

The intersection of the top surface with the constrained space in the center represents the solution

Another simple problem can be defined by the constraints

x₁² − x₂² + x₃² ≤ 2

x₁² + x₂² + x₃² ≤ 10

with an objective function to be maximized

f(x) = x₁x₂ + x₂x₃

where x = (x₁, x₂, x₃).

References

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External links

Software

AIMMS Optimization Modeling AIMMS — includes nonlinear presolve and multi start algorithm;
AMPL solver software - free to students (GUI available)
GAMS General Algebraic Modeling System – free student version available
OPTIMJ A Java language extension for optimization.

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