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FunctionConvexity

FunctionConvexity[f,{x₁,x₂,…}]

finds the convexity of the function f with variables x₁,x₂,… over the reals.

FunctionConvexity[{f,cons},{x₁,x₂,…}]

finds the convexity when variables are restricted by the constraints cons representing a convex region.

Details and Options

Convexity is also known as convex, concave, strictly convex and strictly concave.
By default, the following definitions are used:

		+1	convex, i.e. for all and all and
		0	affine , i.e. for all and all and
		-1	concave, i.e. for all and all and
		Indeterminate	neither convex nor concave

The affine function is both convex and concave.
With the setting StrictInequalitiesTrue, the following definitions are used:

		+1	strictly convex, i.e. for all and all and with
		-1	strictly concave, i.e. for all and all and and
		Indeterminate	neither strictly convex nor strictly concave

The function should be a real-valued function for all real that satisfy the constraints cons.
cons can contain equations, inequalities or logical combinations of these representing a convex region.
The following options can be given:

Assumptions	$Assumptions	assumptions on parameters
GenerateConditions	Automatic	whether to generate conditions on parameters
PerformanceGoal	$PerformanceGoal	whether to prioritize speed or quality
StrictInequalities	False	whether to require strict convexity

Possible settings for GenerateConditions include:
Automatic nongeneric conditions only

True all conditions

False no conditions

None return unevaluated if conditions are needed
Possible settings for PerformanceGoal are "Speed" and "Quality".

Examples

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Basic Examples (3)

Find the convexity of a univariate function:

Find the convexity of a multivariate function:

Find the convexity of a function with variables restricted by constraints:

Scope (7)

Univariate functions:

A function that is not real valued has Indeterminate convexity:

The function is real valued and concave for positive :

Univariate functions with constraints on the variable:

The strict convexity of a function:

is convex, but not strictly convex. $x^2 TemplateBox[{x}, Abs]$ is strictly convex:

Multivariate functions:

Multivariate functions with constraints on variables:

In a different region, the same function is convex:

Functions with symbolic parameters:

Options (5)

Assumptions (1)

FunctionConvexity gives a conditional answer here:

Check convexity for other values of :

GenerateConditions (2)

By default, FunctionConvexity may generate conditions on symbolic parameters:

With GenerateConditionsNone, FunctionConvexity fails instead of giving a conditional result:

This returns a conditionally valid result without stating the condition:

By default, all conditions are reported:

With GenerateConditionsAutomatic, conditions that are generically true are not reported:

PerformanceGoal (1)

Use PerformanceGoal to avoid potentially expensive computations:

The default setting uses all available techniques to try to produce a result:

StrictInequalities (1)

By default, FunctionConvexity computes the non-strict convexity:

With StrictInequalitiesTrue, FunctionConvexity computes the strict sign:

is convex, but not strictly convex. is strictly convex:

Applications (17)

Basic Applications (8)

Check the convexity of :

The segment connecting any two points on the graph lies above the graph:

Check the convexity of :

The segment connecting any two points on the graph lies below the graph:

Check the convexity of :

is neither a convex function nor a concave function:

Show that restricted to is a strictly concave function:

is convex, but not strictly convex:

restricted to positive reals is an affine function:

is convex for , but not strictly convex:

The sum of functions with convexity has convexity :

The negation of a convex function is concave:

The maximum of convex functions is convex:

Affine functions are both convex and concave, hence their maximum is convex:

Minimum of affine functions is concave:

A quadratic is convex iff is positive semidefinite:

Calculus (2)

If is non-decreasing, then is a convex function of :

The derivative of a convex function is non-decreasing:

The second derivative of a convex function is non-negative:

Geometry (4)

If is a convex function $TemplateBox[{}, Reals]^n->TemplateBox[{}, Reals]$ , then the region ${x in TemplateBox[{}, Reals]^n|f(x)<=c}$ is convex:

Use ConvexRegionQ to verify that is a convex region:

If is a concave function $TemplateBox[{}, Reals]^n->TemplateBox[{}, Reals]$ , then the region ${x in TemplateBox[{}, Reals]^n|f(x)>=c}$ is convex:

Use ConvexRegionQ to verify that is a convex region:

If is a convex function, then the epigraph is a convex set:

Use ConvexRegionQ to verify that is a convex region:

If is a concave function, then the hypograph is a convex set:

The region is convex:

Optimization (3)

A local minimum of a convex function is a global minimum:

The set of local and global minima is the non-positive half-line:

A strictly convex function has at most one local minimum:

A strictly convex function may have no local minima:

Properties & Relations (2)

Sum and Max of convex functions are convex:

The second derivative of a convex function is non-negative:

Use D to compute the derivative:

Use FunctionSign to verify that the derivative is non-negative:

Plot the function and the derivative:

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FunctionConvexity

Details and Options

Examples

Basic Examples (3)

Scope (7)