FindArgMin

FindArgMin[f,x]

gives the position xmin of a local minimum of f.

FindArgMin[f,{x,x0}]

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

FindArgMin[f,{{x,x0},{y,y0},…}]

gives the position {xmin,ymin,…} of a local minimum of a function of several variables.

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

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

FindArgMin[{f,cons},{x,y,…}]

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

Details and Options

  • FindArgMin[…,{x,y,…}] is effectively equivalent to {x,y,…}/.Last[FindMinimum[…,{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
  • FindArgMin first localizes the values of all variables, then evaluates f with the variables being symbolic, and then repeatedly evaluates the result numerically.
  • FindArgMin has attribute HoldAll, and effectively uses Block to localize variables.
  • FindArgMin[f,{x,x0,x1}] searches for a local minimum in f using x0 and x1 as the first two values of x, avoiding the use of derivatives.
  • FindArgMin[f,{x,x0,xmin,xmax}] searches for a local minimum, 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 FindArgMin may correspond only to local, but not global, minima.
  • 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.
  • FindArgMin takes the same options as FindMinimum.

    • 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 minimum:

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

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

Find a minimizer point in a geometric region:

Plot it:

Scope  (12)

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

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

Location of a local minimum 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 minimum in a region:

Plot it:

Find the minimum distance between two regions:

Plot it:

Find the minimum such that the triangle and ellipse still intersect:

Plot it:

Find the disk of minimum radius that contains the given three points:

Plot it:

Using Circumsphere gives the same result directly:

Use to specify that is a vector in :

Find the minimum distance between two regions:

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 minimum:

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:

NMinimize also uses a range of direct search methods:

StepMonitor  (1)

Steps taken by FindArgMin in finding the minimum of a function:

WorkingPrecision  (1)

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

Properties & Relations  (1)

FindMinimum gives both the value of the minimum and the minimizer point:

FindArgMin gives the location of the minimum:

FindMinValue gives the value at the minimum:

Possible Issues  (4)

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

If the minimum 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), FindArgMin, Wolfram Language function, https://reference.wolfram.com/language/ref/FindArgMin.html (updated 2014).

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2024_findargmin, author="Wolfram Research", title="{FindArgMin}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/FindArgMin.html}", note=[Accessed: 22-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_findargmin, organization={Wolfram Research}, title={FindArgMin}, year={2014}, url={https://reference.wolfram.com/language/ref/FindArgMin.html}, note=[Accessed: 22-December-2024 ]}