Minimize over the unconstrained reals:

If the single variable is not given in a list, the result is a value at which the minimum is attained:

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:

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:

Piecewise functions:

Unconstrained problems solvable using function property information:

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 linear optimization:

Coordinates of the minimizer point are continuous functions of parameters:

Parametric quadratic optimization:

Coordinates of the minimizer point are continuous functions of parameters:

Unconstrained parametric polynomial minimization:

Constrained parametric polynomial minimization:

Univariate problems:

Integer linear programming:

Polynomial minimization over the integers:

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:

Finding an exact minimum point can take a long time:

With WorkingPrecision->100, you get an approximate minimum point:

Find the lengths of sides of a unit area rectangle with minimal perimeter:

Find the lengths of sides of a unit area triangle with minimal perimeter:

The minimal perimeter triangle is equilateral:

Find a point on a parabola closest to its axis:

Assuming a particular relationship between the and parameters:

The point q in a region ℛ that is nearest to a given point p is given by ArgMin[Norm[p-q],q∈ℛ]. Find the nearest point in Disk[] to {1,1}:

Plot it:

Find the nearest point to {1,2} in the standard unit simplex Simplex[2]:

Plot it:

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

Plot it:

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

Plot it:

The the nearest points p∈ and q∈ can be found through ArgMin[Norm[p-q],{p∈,q∈}]. Find the nearest points in Disk[{0,0}] and Rectangle[{3,3}]:

The nearest distance:

Plot it:

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

The nearest distance:

Plot it:

If ℛ⊆ⁿ is a region that is full dimensional, then the Chebyshev center is the point p∈ℛ that minimizes SignedRegionDistance[ℛ,p], i.e. the negation of the distance to the complement region. Find the Chebyshev center for Disk[]:

Find the Chebyshev center for Rectangle[]: