MinValue

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`MinValue`

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MinValue[f,x]

gives the minimum value of f with respect to x.

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MinValue[f,{x,y,…}]

gives the exact minimum value of f with respect to x, y, ….

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MinValue[{f,cons},{x,y,…}]

gives the minimum value of f subject to the constraints cons.

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MinValue[…,x∈rdom]

constrains x to be in the region or domain rdom.

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MinValue[…,…,dom]

constrains variables to the domain dom, typically Reals or Integers.

Details and Options

MinValue is also known as infimum.
MinValue finds the global minimum of f subject to the constraints given.
MinValue is typically used to find the smallest possible values given constraints. In different areas, this may be called the best strategy, best fit, best configuration and so on.

If f and cons are linear or polynomial, MinValue will always find the global infimum.
The constraints cons can be any logical combination of:

	lhs==rhs	equations
	lhs>rhs, lhs≥rhs, lhs<rhs, lhs≤rhs	inequalities (LessEqual,…)
	lhsrhs, lhsrhs, lhsrhs, lhsrhs	vector inequalities (VectorLessEqual,…)
	Exists[…], ForAll[…]	quantified conditions
	{x,y,…}∈rdom	region or domain specification

MinValue[{f,cons},x∈rdom] is effectively equivalent to MinValue[{f,cons∧x∈rdom},x].
For x∈rdom, the different coordinates can be referred to using Indexed[x,i].
Possible domains rdom include:

	Reals	real scalar variable
	Integers	integer scalar variable
	Vectors[n,dom]	vector variable in
	Matrices[{m,n},dom]	matrix variable in
	ℛ	vector variable restricted to the geometric region

By default, all variables are assumed to be real.
MinValue will return exact results if given exact input. With approximate input, it automatically calls NMinValue.
MinValue will return the following forms:
f_min finite minimum

∞ infeasible, i.e. the constraint set is empty

-∞ unbounded, i.e. the values of f can be arbitrarily small
MinValue gives the infimum of values of f. It may not be attained for any values of x, y, ….
N[MinValue[…]] calls NMinValue for optimization problems that cannot be solved symbolically.

Examples

open allclose all

Basic Examples (5)Summary of the most common use cases

Find the minimum value of a univariate function:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-faqf4h

Out[1]=1

Find the minimum value of a multivariate function:

In[1]:=1

✖

https://wolfram.com/xid/0j4y6y3u-b6nuhh

Out[1]=1

Find the minimum value of a function subject to constraints:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-fcm1i7

Out[1]=1

Find the minimum value as a function of parameters:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-65b19

Out[1]=1

Find the minimum value of a function over a geometric region:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-ewyjra

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0j4y6y3u-dg5h30

Out[2]=2

Scope (36)Survey of the scope of standard use cases

Basic Uses (7)

Minimize over the unconstrained reals:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-86qsf

Out[1]=1

Minimize subject to constraints :

In[1]:=1

✖

https://wolfram.com/xid/0j4y6y3u-edj208

Out[1]=1

Constraints may involve arbitrary logical combinations:

In[1]:=1

✖

https://wolfram.com/xid/0j4y6y3u-dtv7r7

Out[1]=1

An unbounded problem:

In[1]:=1

✖

https://wolfram.com/xid/0j4y6y3u-pjuvi

Out[1]=1

An infeasible problem:

In[1]:=1

✖

https://wolfram.com/xid/0j4y6y3u-jajckq

Out[1]=1

The infimum value may not be attained:

In[1]:=1

✖

https://wolfram.com/xid/0j4y6y3u-fbfbh6

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0j4y6y3u-cgiu1o

Out[2]=2

Use a vector variable and a vector inequality:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-lu8fwi

Out[1]=1

Univariate Problems (7)

Unconstrained univariate polynomial minimization:

In[1]:=1

✖

https://wolfram.com/xid/0j4y6y3u-dhud3y

Out[1]=1

Constrained univariate polynomial minimization:

In[1]:=1

✖

https://wolfram.com/xid/0j4y6y3u-bp80ai

Out[1]=1

Exp-log functions:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-or0am7

Out[1]=1

Analytic functions over bounded constraints:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-cm5gkd

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0j4y6y3u-b1wc7x

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0j4y6y3u-z7foj

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0j4y6y3u-jpja8l

Out[4]=4

Periodic functions:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-ch1ske

Out[1]=1

Combination of trigonometric functions with commensurable periods:

In[2]:=2

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https://wolfram.com/xid/0j4y6y3u-pbsb7r

Out[2]=2

Combination of periodic functions with incommensurable periods:

In[3]:=3

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https://wolfram.com/xid/0j4y6y3u-bqws15

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0j4y6y3u-3ogx

Out[4]=4

Piecewise functions:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-e9jn5u

Out[1]=1

Unconstrained problems solvable using function property information:

In[1]:=1

✖

https://wolfram.com/xid/0j4y6y3u-es0ujg

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0j4y6y3u-jxi0j

Out[2]=2

Multivariate Problems (9)

Multivariate linear constrained minimization:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-cemla

Out[1]=1

Linear-fractional constrained minimization:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-lr473t

Out[1]=1

Unconstrained polynomial minimization:

In[1]:=1

✖

https://wolfram.com/xid/0j4y6y3u-czwmlc

Out[1]=1

Constrained polynomial optimization can always be solved:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-gzq2vq

Out[1]=1

The minimum value may not be attained:

In[2]:=2

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https://wolfram.com/xid/0j4y6y3u-byn0ry

Out[2]=2

The objective function may be unbounded:

In[3]:=3

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https://wolfram.com/xid/0j4y6y3u-i7gioe

Out[3]=3

There may be no points satisfying the constraints:

In[4]:=4

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https://wolfram.com/xid/0j4y6y3u-edzan1

Out[4]=4

Quantified polynomial constraints:

In[5]:=5

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https://wolfram.com/xid/0j4y6y3u-bj2pwq

Out[5]=5

Algebraic minimization:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-dsj2qz

Out[1]=1

Bounded transcendental minimization:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-pc6y7

Out[1]=1

Piecewise minimization:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-cgyyk8

Out[1]=1

Convex minimization:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-81pld6

Out[1]=1

Minimize convex objective function such that is positive semidefinite and :

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-ml87f0

Out[1]=1

Plot the function and the minimum value over the region:

In[2]:=2

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https://wolfram.com/xid/0j4y6y3u-5jo4bk

Out[2]=2

Parametric Problems (4)

Parametric linear optimization:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-ds69uj

Out[1]=1

The minimum value is a continuous function of parameters:

In[2]:=2

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https://wolfram.com/xid/0j4y6y3u-hbs678

Out[2]=2

Parametric quadratic optimization:

In[1]:=1

✖

https://wolfram.com/xid/0j4y6y3u-8l1oto

Out[1]=1

The minimum value is a continuous function of parameters:

In[2]:=2

✖

https://wolfram.com/xid/0j4y6y3u-cgw39

Out[2]=2

Unconstrained parametric polynomial minimization:

In[1]:=1

✖

https://wolfram.com/xid/0j4y6y3u-hvvtn9

Out[1]=1

Constrained parametric polynomial minimization:

In[1]:=1

✖

https://wolfram.com/xid/0j4y6y3u-e7xyfo

Out[1]=1

Optimization over Integers (3)

Univariate problems:

In[1]:=1

✖

https://wolfram.com/xid/0j4y6y3u-ylokw

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0j4y6y3u-0kjmh

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0j4y6y3u-lzeg4t

Out[3]=3

Integer linear programming:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-b4esd5

Out[1]=1

Polynomial minimization over the integers:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-i09s0l

Out[1]=1

Optimization over Regions (6)

Find the minimum value of a function over a geometric region:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-i9nmj9

In[2]:=2

✖

https://wolfram.com/xid/0j4y6y3u-f1e6uo

Out[2]=2

Plot it:

In[3]:=3

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https://wolfram.com/xid/0j4y6y3u-otv084

Out[3]=3

Find the minimum distance between two regions:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-dkte2m

In[2]:=2

✖

https://wolfram.com/xid/0j4y6y3u-la9l1

Out[2]=2

Plot it:

In[3]:=3

✖

https://wolfram.com/xid/0j4y6y3u-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/0j4y6y3u-ho0zb

In[2]:=2

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https://wolfram.com/xid/0j4y6y3u-c5pk5o

Out[2]=2

Plot it:

In[3]:=3

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https://wolfram.com/xid/0j4y6y3u-dy0urb

Out[3]=3

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

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-fxoyak

In[2]:=2

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https://wolfram.com/xid/0j4y6y3u-6vbtl

Out[2]=2

Using Circumsphere gives the same result directly:

In[3]:=3

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https://wolfram.com/xid/0j4y6y3u-ccn5a7

Out[3]=3

Use to specify that is a vector in :

In[1]:=1

✖

https://wolfram.com/xid/0j4y6y3u-e7unye

In[2]:=2

✖

https://wolfram.com/xid/0j4y6y3u-e73rcw

Out[2]=2

Find the minimum distance between two regions:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-kgd51l

In[2]:=2

✖

https://wolfram.com/xid/0j4y6y3u-ktpqdn

Out[2]=2

Plot it:

In[3]:=3

✖

https://wolfram.com/xid/0j4y6y3u-ifh0j0

Out[3]=3

Options (1)Common values & functionality for each option

WorkingPrecision (1)

Finding the exact minimum takes a long time:

In[1]:=1

✖

https://wolfram.com/xid/0j4y6y3u-dt4zl

Out[1]=1

With WorkingPrecision->100, the result is an exact minimum value, but it might be incorrect:

In[2]:=2

✖

https://wolfram.com/xid/0j4y6y3u-c5mizn

Out[2]=2

Applications (9)Sample problems that can be solved with this function

Basic Applications (3)

Find the minimal perimeter among rectangles with a unit area:

In[1]:=1

✖

https://wolfram.com/xid/0j4y6y3u-wtuty

Out[1]=1

Find the minimal perimeter among triangles with a unit area:

In[1]:=1

✖

https://wolfram.com/xid/0j4y6y3u-jayfl1

In[2]:=2

✖

https://wolfram.com/xid/0j4y6y3u-bx6cxm

Out[2]=2

Find the distance to a parabola from a point on its axis:

In[1]:=1

✖

https://wolfram.com/xid/0j4y6y3u-mcgw7a

Out[1]=1

Assuming a particular relationship between the and parameters:

In[2]:=2

✖

https://wolfram.com/xid/0j4y6y3u-h6oc0

Out[2]=2

Geometric Distances (6)

The distance of a point p to a region ℛ is given by MinValue[EuclideanDistance[p,q],q∈ℛ]. Find the distance of {1,1} to the unit Disk[]:

In[1]:=1

✖

https://wolfram.com/xid/0j4y6y3u-bbe5gq

In[4]:=4

✖

https://wolfram.com/xid/0j4y6y3u-ebjkmb

Out[4]=4

Plot it:

In[5]:=5

✖

https://wolfram.com/xid/0j4y6y3u-d8p3tv

Out[5]=5

Find the distance of the point {1,3/4} to the standard unit simplex Simplex[2]:

In[1]:=1

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https://wolfram.com/xid/0j4y6y3u-bi7hqn

In[2]:=2

✖

https://wolfram.com/xid/0j4y6y3u-bfqxai

Out[2]=2

Plot it:

In[3]:=3

✖

https://wolfram.com/xid/0j4y6y3u-hfmqh

Out[3]=3

Find the distance of the point {1,1,1} to the standard unit sphere Sphere[]:

In[1]:=1

✖

https://wolfram.com/xid/0j4y6y3u-dj6j8q

In[2]:=2

✖

https://wolfram.com/xid/0j4y6y3u-g49fxz

Out[2]=2

Plot it:

In[3]:=3

✖

https://wolfram.com/xid/0j4y6y3u-jz2mh

Out[3]=3

Find the distance of the point {-1/3,1/3,1/3} to the standard unit simplex Simplex[3]:

In[1]:=1

✖

https://wolfram.com/xid/0j4y6y3u-jqaty

In[2]:=2

✖

https://wolfram.com/xid/0j4y6y3u-dfalqn

Out[2]=2

Plot it:

In[3]:=3

✖

https://wolfram.com/xid/0j4y6y3u-e5f7z2

Out[3]=3

The distance between regions  and  can be found through MinValue[EuclideanDistance[p,q],{p∈,q∈}]. Find the distance between Disk[{0,0}] and Rectangle[{3,3}]: