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.

Convex sets

A subset of some vector space X is if it satisfies any of the following equivalent conditions: Throughout, will be a map valued in the extended real numbers with a domain that is a convex subset of some vector space. The map is**** a **** if**** holds for any real 0 < r < 1 and any x, y \in X with x \neq y. If this remains true of f when the defining inequality is replaced by the strict inequality then**** f is**** called**** . Convex functions are related to convex sets. Specifically****,**** the function**** f is**** convex**** if**** and only**** if**** its **** is a convex set. 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. The domain**** of**** a function**** is**** denoted by**** **** while its **** is**** the set The function**** is**** called**** **** if**** **** and **** for **** **** Alternatively,**** this**** means that**** there exists**** some**** x in**** the domain**** of**** f at**** which **** and f is**** also**** **** equal to**** -*i**nf**ty**.*** In words, a function is if its domain is not empty, it never takes on the value -\infty, and it also is not identically equal to +\infty. If is a proper convex function then there exist some vector and some such that where x \cdot b denotes the dot product of these vectors.

Convex conjugate

The of an extended real-valued function (not necessarily convex) is the function from the (continuous) dual space X^* of X, and where the brackets denote the canonical duality The of f is the map defined by for every x \in X. If denotes the set of Y-valued functions on X, then the map defined by is called the.

Subdifferential set and the Fenchel-Young inequality

If and x \in X then the is For example, in the important special case where is a norm on X, it can be shown that if then this definition reduces down to: For any x \in X and which is called the. This inequality is an equality (i.e. ) if and only if It is in this way that the subdifferential set is directly related to the convex conjugate

Biconjugate

The of a function is the conjugate of the conjugate, typically written as The biconjugate is useful for showing when strong or weak duality hold (via the perturbation function). For any x \in X, the inequality follows from the. For proper functions, f = f^{**} if and only if f is convex and lower semi-continuous by Fenchel–Moreau theorem.

Convex minimization

A is one of the form

Dual problem

In optimization theory, the 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 and Then given the function we can define the primal problem as finding x such that If there are constraint conditions, these can be built into the function f by letting where I is the indicator function. Then let be a perturbation function such that The with respect to the chosen perturbation function is given by where F^* is the convex conjugate in both variables of F. The duality gap is the difference of the right and left hand sides of the inequality 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:

Lagrange duality

For a convex minimization problem with inequality constraints, the Lagrangian dual problem is where the objective function L(x, u) is the Lagrange dual function defined as follows: