# Minimize

Minimize[f,x]

minimizes f symbolically with respect to x.

Minimize[f,{x,y,}]

minimizes f symbolically with respect to x, y, .

Minimize[{f,cons},{x,y,}]

minimizes f symbolically subject to the constraints cons.

Minimize[,xrdom]

constrains x to be in the region or domain rdom.

Minimize[,,dom]

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

# Details and Options

• Minimize is also known as infimum, symbolic optimization and global optimization (GO).
• Minimize finds the global minimum of f subject to the constraints given.
• Minimize 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.
• Minimize returns a list of the form {fmin,{x->xmin,y->ymin,}}.
• If f and cons are linear or polynomial, Minimize will always find a global minimum.
• The constraints cons can be any logical combination of:
•  lhs==rhs equations lhs>rhs, lhs≥rhs, lhs
• Minimize[{f,cons},xrdom] is effectively equivalent to Minimize[{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.
• Minimize will return exact results if given exact input. With approximate input, it automatically calls NMinimize.
• Minimize will return the following forms:
•  {fmin,{xxmin,…}} finite minimum {∞,{xIndeterminate,…}} infeasible, i.e. the constraint set is empty {-∞,{xxmin,…}} unbounded, i.e. the values of f can be arbitrarily small
• If the minimum is achieved only infinitesimally outside the region defined by the constraints, or only asymptotically, Minimize will return the infimum and the closest specifiable point.
• Even if the same minimum is achieved at several points, only one is returned.
• N[Minimize[]] calls NMinimize for optimization problems that cannot be solved symbolically.
• Minimize[f,x,WorkingPrecision->n] uses n digits of precision while computing a result. »

# Examples

open allclose all

## Basic Examples(5)

Minimize a univariate function:

Minimize a multivariate function:

Minimize a function subject to constraints:

A minimization problem containing parameters:

Minimize a function over a geometric region:

Plot it:

## 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 region and the minimizing point:

### 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)

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

Plot it:

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 solution takes a long time:

With WorkingPrecision->100, you get an exact minimum value, but it might be incorrect:

## Applications(10)

### Basic Applications(3)

Find the minimal perimeter among rectangles with a unit area:

Find the minimal perimeter among triangles with a unit area:

The minimal perimeter triangle is equilateral:

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

Assuming a particular relationship between the and parameters:

### Geometric Distances(6)

The shortest distance of a point in a region to a given point p and a point q realizing the shortest distance is given by Minimize[EuclideanDistance[p,q],q]. Find the shortest distance and the nearest point to {1,1} in the unit Disk[]:

Plot it:

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

Plot it:

Find the shortest distance and the nearest point to {1,1,1} in the standard unit sphere Sphere[]:

Plot it:

Find the shortest distance and the nearest point to {-1/3,1/3,1/3} in the standard unit simplex Simplex[3]:

Plot it:

The nearest points p and q and their distance can be found through Minimize[EuclideanDistance[p,q],{p,q}]. Find the nearest points in Disk[{0,0}] and Rectangle[{3,3}] and the distance between them:

Plot it:

Find the nearest points in Line[{{0,0,0},{1,1,1}}] and Ball[{5,5,0},1] and the distance between them:

Plot it:

### Geometric Centers(1)

If n is a region that is full dimensional, then the Chebyshev center is the center of the largest inscribed ball of . The center and the radius of the largest inscribed ball of can be found through Minimize[SignedRegionDistance[,p], p]. Find the Chebyshev center and the radius of the largest inscribed ball for :

Find the Chebyshev center and the radius of the largest inscribed ball for Triangle[]:

## Properties & Relations(6)

Minimize gives an exact global minimum of the objective function:

NMinimize attempts to find a global minimum numerically, but may find a local minimum:

FindMinimum finds local minima depending on the starting point:

The minimum point satisfies the constraints, unless messages say otherwise:

The given point minimizes the distance from the point {2,}:

When the minimum is not attained, Minimize may give a point on the boundary:

Here the objective function tends to the minimum value when y tends to infinity:

Minimize can solve linear programming problems:

LinearProgramming can be used to solve the same problem given in matrix notation:

This computes the minimum value:

Use RegionDistance and RegionNearest to compute the distance and the nearest point:

Both can be computed using Minimize:

Use RegionBounds to compute the bounding box:

Use Maximize and Minimize to compute the same bounds:

## Possible Issues(1)

Minimize 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 (2003), Minimize, Wolfram Language function, https://reference.wolfram.com/language/ref/Minimize.html (updated 2021).

#### Text

Wolfram Research (2003), Minimize, Wolfram Language function, https://reference.wolfram.com/language/ref/Minimize.html (updated 2021).

#### CMS

Wolfram Language. 2003. "Minimize." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/Minimize.html.

#### APA

Wolfram Language. (2003). Minimize. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Minimize.html

#### BibTeX

@misc{reference.wolfram_2024_minimize, author="Wolfram Research", title="{Minimize}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/Minimize.html}", note=[Accessed: 21-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_minimize, organization={Wolfram Research}, title={Minimize}, year={2021}, url={https://reference.wolfram.com/language/ref/Minimize.html}, note=[Accessed: 21-June-2024 ]}