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 is nonlinear.

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

or

$\min _{x\in X}f(x)$ 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 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 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 or lower than 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 solution cannot be more than a certain percentage better than a solution that has been found. 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.

The Karush-Kuhn-Tucker (KKT) conditions provide the necessary conditions for a solution to be optimal.

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)

See also

• optimization
• curve fitting
• least squares minimization

References

• Avriel, Mordecai (2003). Nonlinear Programming: Analysis and Methods. Dover Publishing. ISBN 0-486-43227-0.

• Bazaraa, Mokhtar S. and Shetty, C. M. (1979). Nonlinear programming. Theory and algorithms. John Wiley & Sons. ISBN 0-471-78610-1.

• Nocedal, Jorge and Wright, Stephen J. (1999). Numerical Optimization. Springer. ISBN 0-387-98793-2.

• Bertsekas, Dimitri P. (1999). Nonlinear Programming: 2nd Edition. Athena Scientific. ISBN 1-886529-00-0.

External links

• Nonlinear programming FAQ
• Mathematical Programming Glossary
• Nonlinear Programming Survey OR/MS Today

Software

• AIMMS Optimization Modeling AIMMS — include nonlinear programming in industry solutions (free trial license available);
• AMPL solver software - free to students (GUI available)
• GAMS General Algebraic Modeling System – free student version available