# 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.

## Convex sets

A convex set is a set CX, for some vector space X, such that for any x, yC and λ ∈ [0, 1] then[1]

$\lambda x + (1 - \lambda)y \in C$.

## Convex functions

A convex function is any extended real-valued function f : XR ∪ {±∞} which satisfies Jensen's inequality, i.e. for any x, yX and any λ ∈ [0, 1] then

$f(\lambda x + (1 - \lambda)y) \leq \lambda f(x) + (1-\lambda) f(y)$.[1]

Equivalently, a convex function is any (extended) real valued function such that its epigraph

$\left\{(x,r) \in X \times \mathbf{R}: f(x) \leq r \right\}$

is a convex set.[1]

## Convex conjugate

The convex conjugate of an extended real-valued (not necessarily convex) function f : XR ∪ {±∞} is f* : X*R ∪ {±∞} where X* is the dual space of X, and[2]:pp.75–79

$f^*(x^*) = \sup_{x \in X} \left \{\langle x^*,x \rangle - f(x) \right \}.$

### Biconjugate

The biconjugate of a function f : XR ∪ {±∞} is the conjugate of the conjugate, typically written as f** : XR ∪ {±∞}. The biconjugate is useful for showing when strong or weak duality hold (via the perturbation function).

For any xX the inequality f**(x) ≤ f(x) follows from the Fenchel–Young inequality. For proper functions, f = f** if and only if f is convex and lower semi-continuous by Fenchel–Moreau theorem.[2]:pp.75–79[3]

## Convex minimization

A convex minimization (primal) problem is one of the form

$\inf_{x \in M} f(x)$

such that f : XR ∪ {±∞} is a convex function and MX is a convex set.

### Dual problem

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 (X, X*) and (Y, Y*). Then given the function f : XR ∪ {+∞}, we can define the primal problem as finding x such that

$\inf_{x \in X} f(x).$

If there are constraint conditions, these can be built into the function f by letting $f = f + I_{\mathrm{constraints}}$ where I is the indicator function. Then let F : X × YR ∪ {±∞} be a perturbation function such that F(x, 0) = f(x).[4]

The dual problem with respect to the chosen perturbation function is given by

$\sup_{y^* \in Y^*} -F^*(0,y^*)$

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[2]:pp. 106–113[4][5]

$\sup_{y^* \in Y^*} -F^*(0,y^*) \le \inf_{x \in X} F(x,0).$

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,

minx f(x) subject to gi(x) ≤ 0 for i = 1, ..., m.

the Lagrangian dual problem is

supu infx L(x, u) subject to ui(x) ≥ 0 for i = 1, ..., m.

where the objective function L(x, u) is the Lagrange dual function defined as follows:

$L(x,u) = f(x) + \sum_{j=1}^m u_j g_j(x)$