Convex function

A function is convex when its value on a line segment is at most the corresponding affine interpolation of endpoint values.

An affine minorant of $f$ is a function $x\mapsto ⟨a,x⟩+b$ that is everywhere at most $f(x)$. Every proper lower-semicontinuous convex function has such a minorant, by separating a point below its closed epigraph. Adding a positive quadratic then yields a coercive function, which establishes existence in proximal minimization.

A proper convex function is both a convex function and a proper extended-real function. For example, the total variation seminorm on a domain is nonnegative, finite at zero, and a supremum of linear functionals, making it proper and convex.

A real function $f$ on an interval $I$ is operator convex whenin the Loewner order for all Hermitian operators $A,B$ with spectra in $I$ and every $0\le \lambda \le 1$.

A function is operator concave when its negative is operator convex. Equivalently, the defining operator-convexity inequality is reversed.

A finite convex function on an open convex subset of a finite-dimensional real vector space is locally Lipschitz, and therefore continuous. In one dimension this follows by bounding its secant slopes between two slightly wider endpoints.

The pointwise maximum of finitely many convex functions is convex because

The softplus function $s(x)=\log (1+e^x)$ is convex because

The absolute value function is convex by the triangle inequality:

A differentiable function is $\alpha$-strongly convex when

A function $f$ on a convex set is strictly convex whenwhenever $x\ne y$ and $0<t<1$.

A strictly convex function has at most one minimizer: if distinct points attained the same minimum, every strict convex combination of them would have a smaller value.

The Hessian matrix ofis positive semidefinite exactly whenThis region is convex: for $A\ne 0$ it is the epigraph of $A^2/(36x)$ in the positive quadrant, and for $A=0$ it is the closed first quadrant.

The perspective of f is g(t,x)=t f(x/t) for positive t and preserves convexity.

A subgradient of a convex function $f$ at $x$ is a vector $g$ such that $f(y)\ge f(x)+g^T(y-x)$ for every $y$ in the domain.

A proper convex function $f$ is minimized at $x$ exactly when $0\in \partial f(x)$. For a differentiable convex term $g$ and another convex term $h$, the subdifferential sum rule gives the condition $0\in \nabla g(x)+\partial h(x)$.

The absolute value function has

A vector is a subgradient when its supporting affine function lies below the convex function.

For a convex function $f$ and an integrable random variable $X$,The inequality is reversed when $f$ is concave.