In mathematics, a real-valued function defined on an interval (or on any convex subset of some vector space) is called convex, concave upwards, concave up or convex cup, if for any two points and in its domain and any , we have

.

In other words, a function is convex if and only if its epigraph (the set of points lying on or above the graph) is a convex set.

Pictorially, a function is called ‘convex’ if the function lies below the straight line segment connecting two points, for any two points in the interval.

A function is called strictly convex if

,

for any and .

A function is said to be concave if is convex.

**First-order condition**

is differentiable if domain of is open and the gradient

exists at each .

1st-order condition: differentiable with convex domain is convex iff

for all .

**Second-order conditions**

is twice differentiable if domain of is open and the Hessian

exists at each .

2nd-order conditions: for twice differentiable with convex domain is convex if and only if

for all . If

for all , then is strictly convex.

**Examples**

* Quadratic function

With

we have

.

Therefore is convex if .

* Least-squares objective

With

we have

.

Therefore, least-squares objective is convex for any .

* Quadratic-over-linear

With we have

and thus

convex for .

* Log-sum-exp

is convex. To this end, note that

To show , we must verify that for all . Indeed,

.

* Geometric mean

on is concave (similar proof as for log-sum-exp).

Source

least-squares objective

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