# Maximize

Maximize[f,x]

maximizes f exactly with respect to x.

Maximize[f,{x,y,}]

maximizes f exactly with respect to x, y, .

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

maximizes f exactly subject to the constraints cons.

Maximize[,xrdom]

constrains x to be in the region or domain rdom.

Maximize[,,dom]

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

# Details and Options   • Maximize is also known as supremum.
• Maximize finds the global maximum of f subject to the constraints given.
• Maximize is typically used to find the largest possible values given constraints. In different areas, this may be called the best strategy, best fit, best configuration and so on.
• Maximize returns a list of the form {fmax,{x->xmax,y->ymax,}}.
• • If f and cons are linear or polynomial, Maximize will always find a global maximum.
• The constraints cons can be any logical combination of:
•  lhs==rhs equations lhs>rhs, lhs≥rhs, lhs
• Maximize[{f,cons},xrdom] is effectively equivalent to Maximize[{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.
• Maximize will return exact results if given exact input. With approximate input, it automatically calls NMaximize.
• Maximize will return the following forms:
•  {fmax,{xxmax,…}} finite maximum {-∞,{xIndeterminate,…}} infeasible, i.e. the constraint set is empty {∞,{xxmax,…}} unbounded, i.e. the values of f can be arbitrarily large
• If the maximum is achieved only infinitesimally outside the region defined by the constraints, or only asymptotically, Maximize will return the supremum and the closest specifiable point.
• Even if the same maximum is achieved at several points, only one is returned.
• N[Maximize[]] calls NMaximize for optimization problems that cannot be solved symbolically.
• Maximize[f,x,WorkingPrecision->n] uses n digits of precision while computing a result. »

# Examples

open allclose all

## Basic Examples(5)

Maximize a univariate function:

Maximize a multivariate function:

Maximize a function subject to constraints:

A maximization problem containing parameters:

Maximize a function over a geometric region:

Plot it:

## Scope(36)

### Basic Uses(7)

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:

### Univariate Problems(7)

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 Problems(9)

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 region and the minimizing point:

### Parametric Problems(4)

Parametric linear optimization:

The maximum value is a continuous function of parameters:

The maximum value is a continuous function of parameters:

Unconstrained parametric polynomial maximization:

Constrained parametric polynomial maximization:

### Optimization over Integers(3)

Univariate problems: Integer linear programming:

Polynomial maximization over the integers:

### Optimization over Regions(6)

Maximize over a region:

Plot it:

Find the maximum distance between points in two regions:

Plot it:

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

Plot it:

Find the maximum for which contains the given three points:

Plot it:

Use to specify that is a vector in :

Find the maximum distance between points in two regions:

Plot it:

## Options(1)

### WorkingPrecision(1)

Finding the exact solution takes a long time:

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

## Applications(13)

### Basic Applications(3)

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:

### Geometric Distances(9)

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

Plot it:

Find the largest distance and the farthest point from {1,3/4} in the standard unit simplex Simplex:

Plot it:

Find the largest distance and the farthest point from {1,1,1} in the standard unit sphere Sphere[]:

Plot it:

Find the largest distance and the farthest point from {-1/3,1/3,1/3} in the standard unit simplex Simplex:

Plot it:

The diameter of a region is the maximum distance between two points in . The diameter and a pair of farthest points in can be computed through Maximize[EuclideanDistance[p,q],{p,q}]. Find the diameter and a pair of farthest points in Circle[]:

Plot it:

Find the diameter and a pair of farthest points in the standard unit simplex Simplex:

Plot it:

Find the diameter and a pair of farthest points in the standard unit cube Cuboid[]:

Plot it:

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

Plot it:

Find the farthest 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 Maximize[-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(4)

Maximize gives an exact global maximum of the objective function:

NMaximize attempts to find a global maximum numerically, but may find a local maximum:

FindMaximum finds local maxima depending on the starting point:

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

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

When the maximum is not attained, Maximize may give a point on the boundary: Here the objective function tends to the maximum value when y tends to infinity: Maximize can solve linear programming problems:

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

This computes the maximum value:

Use RegionBounds to compute the bounding box:

Use Maximize and Minimize to compute the same bounds:

## Possible Issues(1)

Maximize 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: