Maximize over the unconstrained reals:

Maximize subject to constraints :

Constraints may involve arbitrary logical combinations:

An unbounded problem:

An infeasible problem:

The supremum value may not be attained:

Use a vector variable and a vector inequality:

Unconstrained univariate polynomial maximization:

Constrained univariate polynomial maximization:

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 maximization:

Linear-fractional constrained maximization:

Unconstrained polynomial maximization:

Constrained polynomial optimization can always be solved:

The maximum value may not be attained:

The objective function may be unbounded:

There may be no points satisfying the constraints:

Quantified polynomial constraints:

Algebraic maximization:

Bounded transcendental maximization:

Piecewise maximization:

Convex maximization:

Maximize concave objective function such that is positive semidefinite and :

Plot the function and the maximum value over the region:

Parametric linear optimization:

The maximum value is a continuous function of parameters:

Parametric quadratic optimization:

The maximum value is a continuous function of parameters:

Unconstrained parametric polynomial maximization:

Constrained parametric polynomial maximization:

Univariate problems:

Integer linear programming:

Polynomial maximization over the integers:

Find the maximum value of a function over a geometric region:

Plot it:

Find the maximum distance between points in two regions:

Find the maximum such that the triangle and ellipse still intersect:

Plot it:

Find the maximum for which contains the given three points:

Use to specify that is a vector in :

Find the maximum distance between points in two regions:

Finding the exact maximum can take a long time:

With WorkingPrecision->200, you get an exact maximum value, but it might be incorrect:

Find the maximal area among rectangles with a unit perimeter:

Find the maximal area among triangles with a unit perimeter:

Find the maximum height reached by a projectile:

Find the maximum range of a projectile:

The infinity norm of a function f[x] is given by MaxValue[{Norm[f[x]],x∈},x] where  is the domain of interest for f[x]. Find the infinity norm of over the interval {-3,3}:

Plot it:

Find the infinity norm for over Rectangle[{-1,-1},{1,1}]:

Plot it:

The largest distance of a point in a region ℛ to a given point p is given by MaxValue[EuclideanDistance[p,q],q∈ℛ]. Find the largest distance of a point in the unit Disk[] to the point {1,1} :

Plot it:

Find the largest distance of a point in the standard unit simplex Simplex[2] to the point {1,3/4}:

Plot it:

Find the largest distance of a point in the standard unit sphere Sphere[] to the point {1,1,1}:

Plot it:

Find the largest distance of a point in the standard unit simplex Simplex[3] to the point {-1/3,1/3,1/3}:

Plot it:

The diameter of a region ℛ is the maximum distance between two points in ℛ. It can be computed through MaxValue[EuclideanDistance[p,q],{p∈ℛ,q∈ℛ}]. Find the diameter of Circle[]:

Find the diameter of the standard unit simplex Simplex[2]:

Find the diameter of the standard unit cube Cuboid[]:

The largest distance of points p∈ and q∈ can be found through MaxValue[EuclideanDistance[p,q],{p∈,q∈}]. Find the largest distance of points in Disk[{0,0}] and Rectangle[{3,3}]:

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