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:

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 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:

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:

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 the exact solution takes a long time:

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

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:

The shortest distance of a point in a region ℛ to a given point p and a point q realizing the shortest distance is given by Minimize[EuclideanDistance[p,q],q∈ℛ]. 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:

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

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