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FindMinimum
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FindMinimum

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searches for a local minimum in f, starting from an automatically selected point.

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FindMinimum[f,{x,x0}]

searches for a local minimum in f, starting from the point x=x0.

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FindMinimum[f,{{x,x0},{y,y0},…}]

searches for a local minimum in a function of several variables.

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

searches for a local minimum subject to the constraints cons.

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FindMinimum[{f,cons},{x,y,…}]

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

Details and Options

  • FindMinimum returns a list of the form {fmin,{x->xmin}}, where fmin is the minimum value of f found, and xmin is the value of x for which it is found.
  • 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.
  • The constraints 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
  • FindMinimum first localizes the values of all variables, then evaluates f with the variables being symbolic, and then repeatedly evaluates the result numerically.
  • FindMinimum has attribute HoldAll, and effectively uses Block to localize variables.
  • FindMinimum[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.
  • FindMinimum[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 FindMinimum 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.
  • The following options can be given:
  • AccuracyGoalAutomaticthe accuracy sought
    EvaluationMonitor Noneexpression to evaluate whenever f is evaluated
    Gradient Automaticthe list of gradient components for f
    MaxIterationsAutomaticmaximum number of iterations to use
    Method Automaticmethod to use
    PrecisionGoalAutomaticthe precision sought
    StepMonitor Noneexpression to evaluate whenever a step is taken
    WorkingPrecision MachinePrecisionthe 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 minimum, and the value of the function at the minimum.
  • FindMinimum continues until either of the goals specified by AccuracyGoal or PrecisionGoal is achieved.
  • Possible settings for Method include "ConjugateGradient", "PrincipalAxis", "LevenbergMarquardt", "Newton", "QuasiNewton", "InteriorPoint", and "LinearProgramming", with the default being Automatic.

Examples

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Basic Examples  (4)Summary of the most common use cases

Find a local minimum, starting the search at :

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-gieqn3
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0tzrw0wi-f0x936
Out[2]=2

Extract the value of x at the local minimum:

In[3]:=3
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https://wolfram.com/xid/0tzrw0wi-rajau
Out[3]=3

Find a local minimum, starting at , subject to constraints :

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-lh99qh
Out[1]=1

Find the minimum of a linear function, subject to linear and integer constraints:

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-t5y4sr
Out[1]=1

Find a minimum of a function over a geometric region:

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-ewyjra
Out[1]=1

Plot it:

In[2]:=2
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https://wolfram.com/xid/0tzrw0wi-bema4a
Out[2]=2

Scope  (12)Survey of the scope of standard use cases

With different starting points, you may get different local minima:

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-bd0eo9
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0tzrw0wi-gn7af6
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-f4rnn
Out[1]=1

Local minimum constrained within a disk:

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-jrsds
Out[1]=1

Starting point does not have to be provided:

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-clcnhv
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-67ordz
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0tzrw0wi-n2rzg5
Out[2]=2

Or constraints can be specified:

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-6q0ub3
Out[1]=1

Find a minimum in a region:

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-i9nmj9
In[2]:=2
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https://wolfram.com/xid/0tzrw0wi-hkqjtu
Out[2]=2

Plot it:

In[3]:=3
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https://wolfram.com/xid/0tzrw0wi-otv084
Out[3]=3

Find the minimum distance between two regions:

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-dkte2m
In[2]:=2
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https://wolfram.com/xid/0tzrw0wi-khhnmt
Out[2]=2

Plot it:

In[3]:=3
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https://wolfram.com/xid/0tzrw0wi-bvatmu
Out[3]=3

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

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-ho0zb
In[2]:=2
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https://wolfram.com/xid/0tzrw0wi-c896h8
Out[2]=2

Plot it:

In[3]:=3
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https://wolfram.com/xid/0tzrw0wi-dy0urb
Out[3]=3

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

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-fxoyak
In[2]:=2
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https://wolfram.com/xid/0tzrw0wi-h8k62r
Out[2]=2

Plot it:

In[3]:=3
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https://wolfram.com/xid/0tzrw0wi-m75xbi
Out[3]=3

Using Circumsphere gives the same result directly:

In[4]:=4
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https://wolfram.com/xid/0tzrw0wi-ccn5a7
Out[4]=4

Use to specify that is a vector in :

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-e7unye
In[2]:=2
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https://wolfram.com/xid/0tzrw0wi-bfw719
Out[2]=2

Find the minimum distance between two regions:

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-kgd51l
In[2]:=2
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https://wolfram.com/xid/0tzrw0wi-cjdzl7
Out[2]=2

Plot it:

In[3]:=3
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https://wolfram.com/xid/0tzrw0wi-ifh0j0
Out[3]=3

Options  (7)Common values & functionality for each option

AccuracyGoal & PrecisionGoal  (2)

This enforces convergence criteria and :

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-05x0hq
Out[1]=1

This enforces convergence criteria and :

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-ykixh5
Out[1]=1

Setting a high WorkingPrecision makes the process convergent:

In[2]:=2
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https://wolfram.com/xid/0tzrw0wi-xk2rz
Out[2]=2

EvaluationMonitor  (1)

Plot convergence to the local minimum:

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-lyje0
Out[1]=1

Gradient  (1)

Use a given gradient; the Hessian is computed automatically:

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-e7607c
Out[1]=1

Supply both gradient and Hessian:

In[2]:=2
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https://wolfram.com/xid/0tzrw0wi-ii49r8
Out[2]=2

Method  (1)

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

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-m77ild
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0tzrw0wi-ptmgl8
Out[2]=2

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

In[3]:=3
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https://wolfram.com/xid/0tzrw0wi-mgtjz6
Out[3]=3

NMinimize also uses a range of direct search methods:

In[4]:=4
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https://wolfram.com/xid/0tzrw0wi-1698y
Out[4]=4

StepMonitor  (1)

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

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-xmz5e
In[2]:=2
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https://wolfram.com/xid/0tzrw0wi-hd3o67
Out[2]=2

WorkingPrecision  (1)

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

In[1]:=1
✖
https://wolfram.com/xid/0tzrw0wi-tjui3t
Out[1]=1

Applications  (3)Sample problems that can be solved with this function

Annual returns (R) of long bonds from S&P 500 from 1973 to 1994:

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-kptjyg

Compute the mean and covariance from the returns:

In[2]:=2
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https://wolfram.com/xid/0tzrw0wi-b6q4i0
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0tzrw0wi-dz9r7w
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0tzrw0wi-bj60z9
Out[4]=4

Minimize the volatility subject to at least 10% return:

In[5]:=5
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https://wolfram.com/xid/0tzrw0wi-l7nsab
Out[5]=5

Find the smallest square that can contain circles of given radii for that do not overlap. Specify the number of circles and the radius of each circle:

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-m2255i

If is the center of circle , then the objective is to minimize . The objective can be transformed so as to minimize and :

In[2]:=2
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https://wolfram.com/xid/0tzrw0wi-t7gqfm

The circles must not overlap:

In[3]:=3
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https://wolfram.com/xid/0tzrw0wi-22nk9i

Collect the variables:

In[4]:=4
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https://wolfram.com/xid/0tzrw0wi-yemsm5

Minimize the objective :

In[5]:=5
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https://wolfram.com/xid/0tzrw0wi-fazut5

The circles are contained in the square :

In[6]:=6
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https://wolfram.com/xid/0tzrw0wi-obfv8i
Out[6]=6

Compute the fraction of square covered by the circles:

In[7]:=7
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https://wolfram.com/xid/0tzrw0wi-40ke7i
Out[7]=7

Find a path through circular obstacles such that the distance between the start and end points is minimized:

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-b7tj3j
Out[1]=1

The path is discretized into different points with distance of between points, where is the trajectory length being minimized:

In[2]:=2
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https://wolfram.com/xid/0tzrw0wi-518i2b

The points cannot be inside the circular objects:

In[3]:=3
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https://wolfram.com/xid/0tzrw0wi-509nls

The start and end points are known:

In[4]:=4
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https://wolfram.com/xid/0tzrw0wi-7yhvw9

Collect the variables:

In[5]:=5
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https://wolfram.com/xid/0tzrw0wi-wofkjz

Minimize the length subject to the constraints:

In[6]:=6
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https://wolfram.com/xid/0tzrw0wi-7o04by

Visualize the result:

In[7]:=7
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https://wolfram.com/xid/0tzrw0wi-8fgbp9
Out[7]=7

Properties & Relations  (2)Properties of the function, and connections to other functions

FindMinimum tries to find a local minimum; NMinimize attempts to find a global minimum:

In[1]:=1
✖
https://wolfram.com/xid/0tzrw0wi-8v0aqv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tzrw0wi-ny4nl7
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0tzrw0wi-dlfdes
Out[3]=3

Minimize finds a global minimum and can work in infinite precision:

In[4]:=4
✖
https://wolfram.com/xid/0tzrw0wi-zkfa6a
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0tzrw0wi-rq075c
Out[5]=5

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

In[1]:=1
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https://wolfram.com/xid/0tzrw0wi-c60bbs
Out[1]=1

FindArgMin gives the location of the minimum:

In[2]:=2
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https://wolfram.com/xid/0tzrw0wi-540j8q
Out[2]=2

FindMinValue gives the value at the minimum:

In[3]:=3
✖
https://wolfram.com/xid/0tzrw0wi-1wr2eh
Out[3]=3

Possible Issues  (6)Common pitfalls and unexpected behavior

With machine-precision arithmetic, even functions with smooth minima may seem bumpy:

In[1]:=1
✖
https://wolfram.com/xid/0tzrw0wi-e724l1
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tzrw0wi-de0ghh
In[3]:=3
✖
https://wolfram.com/xid/0tzrw0wi-dd83h6
Out[3]=3

Going beyond machine precision often avoids such problems:

In[4]:=4
✖
https://wolfram.com/xid/0tzrw0wi-hiyaum
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0tzrw0wi-vy9gjh
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0tzrw0wi-mongm3
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tzrw0wi-wql3uj
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0tzrw0wi-uzrdll
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0tzrw0wi-n09ud0
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tzrw0wi-mx1mby
Out[2]=2

It can be time-consuming to compute functions symbolically:

In[1]:=1
✖
https://wolfram.com/xid/0tzrw0wi-qq24op
In[2]:=2
✖
https://wolfram.com/xid/0tzrw0wi-tptg2r
Out[2]=2

Restricting the function definition prevents symbolic evaluation:

In[3]:=3
✖
https://wolfram.com/xid/0tzrw0wi-ktvle7
In[4]:=4
✖
https://wolfram.com/xid/0tzrw0wi-4ebq4m
Out[4]=4
Wolfram Research (1988), FindMinimum, Wolfram Language function, https://reference.wolfram.com/language/ref/FindMinimum.html (updated 2014).
✖
Wolfram Research (1988), FindMinimum, Wolfram Language function, https://reference.wolfram.com/language/ref/FindMinimum.html (updated 2014).

Text

Wolfram Research (1988), FindMinimum, Wolfram Language function, https://reference.wolfram.com/language/ref/FindMinimum.html (updated 2014).

✖
Wolfram Research (1988), FindMinimum, Wolfram Language function, https://reference.wolfram.com/language/ref/FindMinimum.html (updated 2014).

CMS

Wolfram Language. 1988. "FindMinimum." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/FindMinimum.html.

✖
Wolfram Language. 1988. "FindMinimum." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/FindMinimum.html.

APA

Wolfram Language. (1988). FindMinimum. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FindMinimum.html

✖
Wolfram Language. (1988). FindMinimum. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FindMinimum.html

BibTeX

@misc{reference.wolfram_2025_findminimum, author="Wolfram Research", title="{FindMinimum}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/FindMinimum.html}", note=[Accessed: 09-July-2025 ]}

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@misc{reference.wolfram_2025_findminimum, author="Wolfram Research", title="{FindMinimum}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/FindMinimum.html}", note=[Accessed: 09-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_findminimum, organization={Wolfram Research}, title={FindMinimum}, year={2014}, url={https://reference.wolfram.com/language/ref/FindMinimum.html}, note=[Accessed: 09-July-2025 ]}

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@online{reference.wolfram_2025_findminimum, organization={Wolfram Research}, title={FindMinimum}, year={2014}, url={https://reference.wolfram.com/language/ref/FindMinimum.html}, note=[Accessed: 09-July-2025 ]}