FindArgMax

FindArgMax[f,x]

gives the position xmax of a local maximum of f.

FindArgMax[f,{x,x0}]

gives the position xmax of a local maximum of f, found by a search starting from the point x=x0.

FindArgMax[f,{{x,x0},{y,y0},…}]

gives the position {xmax,ymax,…} of a local maximum of a function of several variables.

FindArgMax[{f,cons},{{x,x0},{y,y0},…}]

gives the position of a local maximum subject to the constraints cons.

FindArgMax[{f,cons},{x,y,…}]

starts from a point within the region defined by the constraints.

Details and Options

  • FindArgMax[…,{x,y,…}] is effectively equivalent to {x,y,…}/.Last[FindMaximum[…,{x,y,…},…].
  • If the starting point for a variable is given as a list, the values of the variable are taken to be lists with the same dimensions.
  • 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
  • FindArgMax first localizes the values of all variables, then evaluates f with the variables being symbolic, and then repeatedly evaluates the result numerically.
  • FindArgMax has attribute HoldAll, and effectively uses Block to localize variables.
  • FindArgMax[f,{x,x0,x1}] searches for a local maximum in f using x0 and x1 as the first two values of x, avoiding the use of derivatives.
  • FindArgMax[f,{x,x0,xmin,xmax}] searches for a local maximum, stopping the search if x ever gets outside the range xmin to xmax.
  • Except when f and cons are both linear, the results found by FindArgMax may correspond only to local, but not global, maxima.
  • By default, all variables are assumed to be real.
  • For linear f and cons, x∈Integers can be used to specify that a variable can take on only integer values.
  • FindArgMax takes the same options as FindMaximum.

    • List of all options

Examples

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Basic Examples  (4)

Find a point {x} at which the univariate function -2x^2-3x+5 has a maximum:

Find a point {x,y} at which the function Sin[x]Sin[2y] has a maximum:

Find a point at which a function is a maximum subject to constraints:

Find a maximizer point for a function over a geometric region:

Plot it:

Scope  (12)

With different starting points, get the locations of different local maxima:

Location of a local maximum of a two-variable function starting from x=2, y=2:

Location of a local maximum constrained within a disk:

Starting point does not have to be provided:

For linear objective and constraints, integer constraints can be imposed:

Or constraints can be specified:

Find a maximum in a region:

Plot it:

Find points in two regions realizing the maximum distance:

Plot it:

Find the maximum such that the rectangle and ellipse still intersect:

Plot it:

Find the maximum for which contains the given three points:

Plot it:

Use to specify that is a vector in :

Find points in two regions realizing the maximum distance:

Plot it:

Options  (7)

AccuracyGoal & PrecisionGoal  (2)

This enforces convergence criteria TemplateBox[{{{x, _, k}, -, {x, ^, *}}}, Norm]<=max(10^(-9),10^(-8)TemplateBox[{{x, _, k}}, Norm]) and TemplateBox[{{del , {f, (, {x, _, k}, )}}}, Norm]<=10^(-9):

This enforces convergence criteria TemplateBox[{{{x, _, k}, -, {x, ^, *}}}, Norm]<=max(10^(-20),10^(-18)TemplateBox[{{x, _, k}}, Norm]) and TemplateBox[{{del , {f, (, {x, _, k}, )}}}, Norm]<=10^(-20):

Setting a high WorkingPrecision makes the process convergent:

EvaluationMonitor  (1)

Plot convergence to the local maximum:

Gradient  (1)

Use a given gradient; the Hessian is computed automatically:

Supply both gradient and Hessian:

Method  (1)

In this case the default derivative-based methods have difficulties:

Direct search methods that do not require derivatives can be helpful in these cases:

NMaximize also uses a range of direct search methods:

StepMonitor  (1)

Steps taken by FindArgMax in finding the maximum of a function:

WorkingPrecision  (1)

Set the working precision to ; by default AccuracyGoal and PrecisionGoal are set to :

Properties & Relations  (1)

FindMaximum gives both the value of the maximum and the maximizer point:

FindArgMax gives the location of the maximum:

FindMaxValue gives the value at the maximum:

Possible Issues  (4)

If the constraint region is empty, the algorithm will not converge:

If the maximum value is not finite, the algorithm will not converge:

Integer linear programming algorithm is only available for machine-number problems:

Sometimes providing a suitable starting point can help the algorithm to converge:

Wolfram Research (2008), FindArgMax, Wolfram Language function, https://reference.wolfram.com/language/ref/FindArgMax.html (updated 2014).

Text

Wolfram Research (2008), FindArgMax, Wolfram Language function, https://reference.wolfram.com/language/ref/FindArgMax.html (updated 2014).

CMS

Wolfram Language. 2008. "FindArgMax." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/FindArgMax.html.

APA

Wolfram Language. (2008). FindArgMax. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FindArgMax.html

BibTeX

@misc{reference.wolfram_2024_findargmax, author="Wolfram Research", title="{FindArgMax}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/FindArgMax.html}", note=[Accessed: 21-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_findargmax, organization={Wolfram Research}, title={FindArgMax}, year={2014}, url={https://reference.wolfram.com/language/ref/FindArgMax.html}, note=[Accessed: 21-December-2024 ]}