# 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 , 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. 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 , 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 , 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 , 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 , then the region is convex:

Use ConvexRegionQ to verify that is a convex region:

If is a concave function , then the region 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: