MinValue

MinValue[f,x]

gives the minimum value of f with respect to x.

MinValue[f,{x,y,…}]

gives the exact 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∈rdom]

constrains x to be in the region or domain rdom.

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==rhsequations
    lhs>rhs, lhs≥rhs, lhs<rhs, lhs≤rhsinequalities (LessEqual,…)
    lhsrhs, lhsrhs, lhsrhs, lhsrhsvector inequalities (VectorLessEqual,…)
    Exists[…], ForAll[…]quantified conditions
    {x,y,…}∈rdomregion 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:
  • Realsreal scalar variable
    Integersinteger scalar variable
    Vectors[n,dom]vector variable in
    Matrices[{m,n},dom]matrix variable in
    ℛvector 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:
  • fminfinite 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

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Basic Examples  (5)

Find the minimum value of a univariate function:

Find the minimum value of a multivariate function:

Find the minimum value of a function subject to constraints:

Find the minimum value as a function of parameters:

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

Scope  (36)

Basic Uses  (7)

Minimize over the unconstrained reals:

Minimize subject to constraints :

Constraints may involve arbitrary logical combinations:

An unbounded problem:

An infeasible problem:

The infimum value may not be attained:

Use a vector variable and a vector inequality:

Univariate Problems  (7)

Unconstrained univariate polynomial minimization:

Constrained univariate polynomial minimization:

Exp-log functions:

Analytic functions over bounded constraints:

Periodic functions:

Combination of trigonometric functions with commensurable periods:

Combination of periodic functions with incommensurable periods:

Piecewise functions:

Unconstrained problems solvable using function property information:

Multivariate Problems  (9)

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:

Quantified polynomial constraints:

Algebraic minimization:

Bounded transcendental minimization:

Piecewise minimization:

Convex minimization:

Minimize convex objective function such that is positive semidefinite and :

Plot the function and the minimum value over the region:

Parametric Problems  (4)

Parametric linear optimization:

The minimum value is a continuous function of parameters:

Parametric quadratic optimization:

The minimum value is a continuous function of parameters:

Unconstrained parametric polynomial minimization:

Constrained parametric polynomial minimization:

Optimization over Integers  (3)

Univariate problems:

Integer linear programming:

Polynomial minimization over the integers:

Optimization over Regions  (6)

Find the minimum value of a function over a geometric 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 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 :

Find the minimum distance between two regions:

Plot it:

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)

Find the minimal perimeter among rectangles with a unit area:

Find the minimal perimeter among triangles with a unit area:

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

Assuming a particular relationship between the and parameters:

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[]:

Plot it:

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

Plot it:

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

Plot it:

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

Plot it:

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:

Use MaxValue and MinValue to compute the same bounds:

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:

Wolfram Research (2008), MinValue, Wolfram Language function, https://reference.wolfram.com/language/ref/MinValue.html (updated 2021).

Text

Wolfram Research (2008), MinValue, Wolfram Language function, https://reference.wolfram.com/language/ref/MinValue.html (updated 2021).

CMS

Wolfram Language. 2008. "MinValue." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. 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

BibTeX

@misc{reference.wolfram_2022_minvalue, author="Wolfram Research", title="{MinValue}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/MinValue.html}", note=[Accessed: 02-June-2023 ]}

BibLaTeX

@online{reference.wolfram_2022_minvalue, organization={Wolfram Research}, title={MinValue}, year={2021}, url={https://reference.wolfram.com/language/ref/MinValue.html}, note=[Accessed: 02-June-2023 ]}