NMaximize

NMaximize[f,x]

maximizes f numerically with respect to x.

NMaximize[f,{x,y,…}]

maximizes f numerically with respect to x, y, ….

NMaximize[{f,cons},{x,y,…}]

maximizes f numerically subject to the constraints cons.

NMaximize[…,x∈reg]

constrains x to be in the region reg.

Details and Options

  • NMaximize returns a list of the form {fmax,{x->xmax,y->ymax,…}}.
  • cons can contain equations, inequalities or logical combinations of these.
  • The constraints cons can be any logical combination of:
  • lhs==rhsequations
    lhs>rhs or lhs>=rhs inequalities
    {x,y,…}∈regregion specification
  • NMaximize[{f,cons},x∈reg] is effectively equivalent to NMaximize[{f,cons∧x∈reg},x].
  • For x∈reg, the different coordinates can be referred to using Indexed[x,i].
  • NMaximize always attempts to find a global maximum of f subject to the constraints given.
  • By default, all variables are assumed to be real.
  • x∈Integers can be used to specify that a variable can take on only integer values.
  • If f and cons are linear, NMaximize can always find global maxima, over both real and integer values.
  • Otherwise, NMaximize may sometimes find only a local maximum.
  • If NMaximize determines that the constraints cannot be satisfied, it returns {-Infinity,{x->Indeterminate,…}}.
  • The following options can be given:
  • AccuracyGoalAutomaticnumber of digits of final accuracy sought
    EvaluationMonitorNoneexpression to evaluate whenever f is evaluated
    MaxIterationsAutomaticmaximum number of iterations to use
    MethodAutomaticmethod to use
    PrecisionGoalAutomaticnumber of digits of final precision sought
    StepMonitorNoneexpression to evaluate whenever a step is taken
    WorkingPrecisionMachinePrecisionthe precision used in internal computations
  • The settings for AccuracyGoal and PrecisionGoal specify the number of digits to seek in both the value of the position of the maximum, and the value of the function at the maximum.
  • NMaximize continues until either of the goals specified by AccuracyGoal or PrecisionGoal is achieved.
  • Possible settings for the Method option include "NelderMead", "DifferentialEvolution", "SimulatedAnnealing", and "RandomSearch".

Examples

open all close all

Basic Examples  (3)

Find the global maximum of an unconstrained problem:

In[1]:=
Click for copyable input
Out[1]=

Extract the maximizing argument:

In[2]:=
Click for copyable input
Out[2]=

Find the global maximum of problems with constraints:

In[1]:=
Click for copyable input
Out[1]=
In[2]:=
Click for copyable input
Out[2]=

Maximize a function over a geometric region:

In[1]:=
Click for copyable input
Out[1]=

Plot it:

In[2]:=
Click for copyable input
Out[2]=

Scope  (9)

Options  (6)

Properties & Relations  (1)

Possible Issues  (3)

Introduced in 2003
(5.0)
|
Updated in 2014
(10.0)