MinValue
MinValue[f,x]
gives the minimum value of f with respect to x.
MinValue[f,{x,y,…}]
gives the minimum value of f with respect to x, y, ….
MinValue[{f,cons},{x,y,…}]
gives the minimum value of f subject to the constraints cons.
MinValue[…,x∈reg]
constrains x to be in the region reg.
Details and Options

- MinValue[…] is effectively equivalent to First[Minimize[…]].
- MinValue gives the infimum of values of f. It may not be attained for any values of x, y, ….
- cons can contain equations, inequalities or logical combinations of these.
- The constraints cons can be any logical combination of:
-
lhs==rhs equations lhs!=rhs inequations lhs>rhs or lhs>=rhs inequalities {x,y,…}∈reg region specification Exists[x,cond,expr] existential quantifiers - If f and cons are linear or polynomial, MinValue will always find a global minimum.
- MinValue[{f,cons},x∈reg] is effectively equivalent to MinValue[{f,cons∧x∈reg},x].
- For x∈reg, the different coordinates can be referred to using Indexed[x,i].
- MinValue will return exact results if given exact input.
- If MinValue is given an expression containing approximate numbers, it automatically calls NMinValue.
- If no domain is specified, all variables are assumed to be real.
- x∈Integers can be used to specify that a particular variable can take on only integer values.
- If the constraints cannot be satisfied, MinValue returns Infinity.
- N[MinValue[…]] calls NMinValue for optimization problems that cannot be solved symbolically.
Examples
open allclose allBasic Examples (5)
Scope (21)
Unconstrained univariate polynomial minimization:
Constrained univariate polynomial minimization:
Univariate transcendental minimization:
Univariate piecewise minimization:
Multivariate linear constrained minimization:
Linear-fractional constrained minimization:
Unconstrained polynomial minimization:
Constrained polynomial optimization can always be solved:
The minimum value may not be attained:
The objective function may be unbounded:
There may be no points satisfying the constraints:

Bounded transcendental minimization:
Unconstrained parametric minimization:
Constrained parametric minimization:
Polynomial minimization over the integers:
Find the minimum value of a function over a geometric region:
Find the minimum distance between two regions:
Find the minimum such that the triangle and ellipse still intersect:
Find the minimum radius of a disk that contains the given three points:
Using Circumsphere gives the same result directly:
Use to specify that
is a vector in
:
Options (1)
WorkingPrecision (1)
Finding the exact minimum takes a long time:
With WorkingPrecision->100, the result is an exact minimum value, but it might be incorrect:
Applications (9)
Basic Applications (3)
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[]:
Find the distance of the point {1,3/4} to the standard unit simplex Simplex[2]:
Find the distance of the point {1,1,1} to the standard unit sphere Sphere[]:
Find the distance of the point {-1/3,1/3,1/3} to the standard unit simplex Simplex[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}]:
Find the distance between Line[{{0,0,0},{1,1,1}}] and Ball[{5,5,0},1]:
Properties & Relations (5)
Minimize gives both the value of the minimum and the minimizer point:
MinValue gives an exact global minimum value of the objective function:
NMinValue attempts to find a global minimum numerically, but may find a local minimum:
FindMinValue finds local minima depending on the starting point:
MinValue can solve linear programming problems:
LinearProgramming can be used to solve the same problem given in matrix notation:
Use RegionDistance to compute the minimum distance from a point to a region:
Compute the distance using MinValue:
Use RegionBounds to compute the bounding box:
Possible Issues (1)
MinValue requires that all functions present in the input be real valued:
Values for which the equation is satisfied but the square roots are not real are disallowed:
Text
Wolfram Research (2008), MinValue, Wolfram Language function, https://reference.wolfram.com/language/ref/MinValue.html (updated 2014).
BibTeX
BibLaTeX
CMS
Wolfram Language. 2008. "MinValue." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/MinValue.html.
APA
Wolfram Language. (2008). MinValue. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MinValue.html