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 function and the minimum value over the region:

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:

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:

Finding the exact minimum takes a long time:

With WorkingPrecision->100, the result is 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:

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

Assuming a particular relationship between the and parameters:

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]: