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Pseudoconvex function

Type of function

Wikipedia contributorsAdapted for Fathomly · View authors
3 min read
Source updated October 25, 2025

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.

01Formal definition

Consider a differentiable function f:X\subseteq \mathbb {R} ^{n}\rightarrow \mathbb {R}, defined on a (nonempty) convex open set X of the finite-dimensional Euclidean space \mathbb {R} ^{n}. This function is said to be pseudoconvex if the following property holds:

for all x,y\in X:\quad \nabla f(x)\cdot (y-x)\geq 0\Rightarrow f(y)\geq f(x).

Equivalently:

for all x,y\in X:\quad f(y)<f(x)\Rightarrow \nabla f(x)\cdot (y-x)<0.

Here \nabla f is the gradient of f, defined by: \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, in the direction given by the vector v=y-x. This is because, as f is differentiable, this directional derivative is given by:

{\frac {\partial f}{\partial v}}(x)=\nabla f(x)\cdot v=\nabla f(x)\cdot (y-x).
Functions x^3 (quasiconvex but not pseudoconvex) and x^3 + x (pseudoconvex and thus quasiconvex). None of them is convex.
Functions x^3 (quasiconvex but not pseudoconvex) and x^3 + x (pseudoconvex and thus quasiconvex). None of them is convex.

02Properties

Relation to other types of "convexity"

Every convex function is pseudoconvex, but the converse is not true. For example, the function f(x)=x+x^{3} is pseudoconvex but not convex. Similarly, any pseudoconvex function is quasiconvex; but the converse is not true, since the function f(x)=x^{3} is quasiconvex but not pseudoconvex. This can be summarized schematically as:

convex \Rightarrow pseudoconvex \Rightarrow quasiconvex

To see that f(x)=x^{3} is not pseudoconvex, consider its derivative at x=0: f^{\prime }(0)=0. Then, if f(x)=x^{3} was pseudoconvex, we should have:

f^{\prime }(0)(y-0)=0\geq 0\Rightarrow f(y)\geq f(0),\quad \forall \,y\in \mathbb {R} .

In particular it should be true for y=-1. But it is not, as: f(-1)=(-1)^{3}=-1<f(0)=0.

Sufficient optimality condition

For any differentiable function, we have the Fermat's theorem necessary condition of optimality, which states that: if f has a local minimum at x^{*} in an open domain, then x^{*} must be a stationary point of f (that is: \nabla f(x^{*})=0).

Pseudoconvexity is of great interest in the area of optimization, because the converse is also true for any pseudoconvex function. That is: if x^{*} is a stationary point of a pseudoconvex function f, then f has a global minimum at x^{*}. Note also that the result guarantees a global minimum (not only local).

This last result is also true for a convex function, but it is not true for a quasiconvex function. Consider for example the quasiconvex function:

f(x)={\frac {e^{x}}{x^{2}+1}}+{\frac {1}{e^{x}}}.

This function is not pseudoconvex, but it is quasiconvex. Also, the point x=0 is a critical point of f, as f^{\prime }(0)=0. However, f does not have a global minimum at x=0 (not even a local minimum).

Finally, note that a pseudoconvex function may not have any critical point. Take for example the pseudoconvex function: f(x)=x^{3}+x, whose derivative is always positive: f^{\prime }(x)=3x^{2}+1>0,\,\forall \,x\in \mathbb {R}.

Example of a quasiconvex function that is not pseudoconvex. The function has a critical point at , but this is not a minimum.
Example of a quasiconvex function that is not pseudoconvex. The function has a critical point at , but this is not a minimum.

03Examples

An example of a function that is pseudoconvex, but not convex, is: f(x)={\frac {x^{2}}{x^{2}+k}},\,k>0. The figure shows this function for the case where k=0.2. This example may be generalized to two variables as:

f(x)={\frac {x^{2}+y^{2}}{x^{2}+y^{2}+k}},\,k>0.

The previous example may be modified to obtain a function that is not convex, nor pseudoconvex, but is quasiconvex:

f(x)={\frac {|x|^{p}}{|x|^{p}+k}},\,k>0,\,p\in (0,1).

The figure shows this function for the case where k=0.5,p=0.6. As can be seen, this function is not convex because of the concavity, and it is not pseudoconvex because it is not differentiable at x=0.

Pseudoconvex function that is not convex.
Pseudoconvex function that is not convex.

04Generalization to nondifferentiable functions

The notion of pseudoconvexity can be generalized to nondifferentiable functions as follows. Given any function f:X\rightarrow \mathbb {R}, we can define the upper Dini derivative of f by:

f^{+}(x,u)=\limsup _{h\to 0^{+}}{\frac {f(x+hu)-f(x)}{h}};

where u is any unit vector. The function is said to be pseudoconvex if it is increasing in any direction where the upper Dini derivative is positive. More precisely, this is characterized in terms of the subdifferential \partial f as follows:

For all x,y\in X: if x^{*}\in \partial f(x) is such that \langle x^{*},y-x\rangle \geq 0, then f(x)\leq f(z), for all z\in [x,y];

where [x,y] denotes the line segment adjoining x and y.

Quasiconvex function that is not convex, nor pseudoconvex.
Quasiconvex function that is not convex, nor pseudoconvex.
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Sources and credits

This article is adapted from the Wikipedia article “Pseudoconvex function”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Pseudoconvexity

In mathematics, more precisely in the theory of functions of several complex variables, a pseudoconvex set is a special type of open set in the n-dimensional complex space Cn. Pseudoconvex sets are important, as they allow for classification of domains of holomorphy.

Convex function

In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph of the function between the two points. Equivalently, a function is convex if its epigraph is a convex set.

Quasiconvex function

In mathematics, a quasiconvex function is a real-valued function defined on a convex subset of a real vector space, such that for any real number y, the set of points on which the function value is at most y is a convex set. In other words, the inverse image of any set of the form {\displaystyle (-\infty ,y)} is a convex set.