Convex analysis

Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.

is convex if it satisfies any of the following equivalent conditions: Throughout,

will be a map valued in the extended real numbers

that is a convex subset of some vector space.

is a convex function if holds for any real

is called strictly convex.

[2] The epigraphs of extended real-valued functions play a role in convex analysis that is analogous to the role played by graphs of real-valued function in real analysis.

Specifically, the epigraph of an extended real-valued function provides geometric intuition that can be used to help formula or prove conjectures.

while its effective domain is the set[2] The function

[2] Alternatively, this means that there exists some

In words, a function is proper if its domain is not empty, it never takes on the value

is a proper convex function then there exist some vector

denotes the dot product of these vectors.

The convex conjugate of an extended real-valued function

(not necessarily convex) is the function

from the (continuous) dual space

is called the Legendre-Fenchel transform.

then the subdifferential set is For example, in the important special case where

is directly related to the convex conjugate

The biconjugate is useful for showing when strong or weak duality hold (via the perturbation function).

is convex and lower semi-continuous by Fenchel–Moreau theorem.

[3][4] A convex minimization (primal) problem is one of the form In optimization theory, the duality principle states that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem.

In general given two dual pairs separated locally convex spaces

we can define the primal problem as finding

such that If there are constraint conditions, these can be built into the function

[5] The dual problem with respect to the chosen perturbation function is given by where

The duality gap is the difference of the right and left hand sides of the inequality[6][5][7] This principle is the same as weak duality.

If the two sides are equal to each other, then the problem is said to satisfy strong duality.

There are many conditions for strong duality to hold such as: For a convex minimization problem with inequality constraints, the Lagrangian dual problem is where the objective function

is the Lagrange dual function defined as follows:

A 3-dimensional convex polytope. Convex analysis includes not only the study of convex subsets of Euclidean spaces but also the study of convex functions on abstract spaces.
Convex function on an interval.
A function (in black) is convex if and only if its epigraph, which is the region above its graph (in green), is a convex set .
A graph of the bivariate convex function