Pseudoconvex function

In convex analysis and the calculus of variations, both branches of mathematics, a pseudoconvex function is a function that behaves like a convex function with respect to finding its local minima, but need not actually be convex. Informally, a differentiable function is pseudoconvex if it is increasing in any direction where it has a positive directional derivative. The property must hold in all of the function domain, and not only for nearby points.

Formal definition

Consider a differentiable function

f

:

X

⊆

R

n

→

R

{\displaystyle f:X\subseteq \mathbb {R} ^{n}\rightarrow \mathbb {R} }

, defined on a (nonempty) convex open set

X

{\displaystyle X}

of the finite-dimensional Euclidean space

R

n

{\displaystyle \mathbb {R} ^{n}}

. This function is said to be pseudoconvex if the following property holds:

for all

x

,

y

∈

X

:

∇

f

(

x

)

⋅

(

y

−

x

)

≥

0

⇒

f

(

y

)

≥

f

(

x

)

.

{\displaystyle x,y\in X:\quad \nabla f(x)\cdot (y-x)\geq 0\Rightarrow f(y)\geq f(x).}

Equivalently:

for all

x

,

y

∈

X

:

f

(

y

)

<

f

(

x

)

⇒

∇

f

(

x

)

⋅

(

y

−

x

)

<

0.

{\displaystyle x,y\in X:\quad f(y)<f(x)\Rightarrow \nabla f(x)\cdot (y-x)<0.}

Here

∇

f

{\displaystyle \nabla f}

is the gradient of

f

{\displaystyle f}

, defined by:

∇

f

=

(

∂

f

∂

x

1

,

…

,

∂

f

∂

x

n

)

.

{\displaystyle \nabla f=\left({\frac {\partial f}{\partial x_{1}}},\dots ,{\frac {\partial f}{\partial x_{n}}}\right).}

Note that the definition may also be stated in terms of the directional derivative of

f

{\displaystyle f}

, in the direction given by the vector

v

=

y

−

x

{\displaystyle v=y-x}

. This is because, as

f

{\displaystyle f}

is differentiable, this directional derivative is given by:

∂

f

∂

v

(

x

)

=

∇

f

(

x

)

⋅

v

=

∇

f

(

x

)

⋅

(

y

−

x

)

.

{\displaystyle {\frac {\partial f}{\partial v}}(x)=\nabla f(x)\cdot v=\nabla f(x)\cdot (y-x).}

Properties

Relation to other types of "convexity"

Every convex function is pseudoconvex, but the converse is not true.