# 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[,xrdom]

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==rhs equations lhs>rhs, lhs≥rhs, lhs
• MinValue[{f,cons},xrdom] is effectively equivalent to MinValue[{f,consxrdom},x].
• For xrdom, 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:
•  fmin 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

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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:

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 is given by MinValue[EuclideanDistance[p,q],q]. 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:

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:

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: