particular the subfields of convex analysis and optimization, a proper convex function is an extended real-valued convex function with a non-empty domain...

the function f {\displaystyle f} is closed. This definition is valid for any function, but most used for convex functions. A proper convex function is...

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

optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known...

the theory of convex functions named after Werner Fenchel. Let ƒ be a proper convex function on Rn and let g be a proper concave function on Rn. Then,...

In convex analysis, Danskin's theorem is a theorem which provides information about the derivatives of a function of the form f ( x ) = max z ∈ Z ϕ ( x...

manageable condition than a holomorphically convex. The subharmonic function looks like a kind of convex function, so it was named by Levi as a pseudoconvex...

combinations with positive coefficients. It follows that convex cones are convex sets. The definition of a convex cone makes sense in a vector space over any ordered...

In convex analysis, a branch of mathematics, the effective domain extends of the domain of a function defined for functions that take values in the extended...

these functions. Epigraphs serve this same purpose in the fields of convex analysis and variational analysis, in which the primary focus is on convex functions...

In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined...

caveat: many terms in Riemannian and metric geometry, such as convex function, convex set and others, do not have exactly the same meaning as in general...

exponential functions Inverse function Convex function, Concave function Singular function Harmonic function Weakly harmonic function Proper convex function Rational...

are reversed, so that we say the cost game is convex if the characteristic function is submodular. Convex cooperative games have many nice properties:...

and a convex class F {\displaystyle {\mathcal {F}}} of probability measures on ( Ω , A ) {\displaystyle (\Omega ,{\mathcal {A}})} . A function defined...

graph. Closed convex function - a convex function all of whose sublevel sets are closed sets. Proper convex function - a convex function whose effective...

H {\displaystyle H} indicates the Heaviside step function. However, this loss function is non-convex and non-smooth, and solving for the optimal solution...

closed under convex combinations, but not compositions. This class includes proximal mappings of proper, convex, lower-semicontinuous functions, hence it...

characteristic function in convex analysis, which is defined as if using the reciprocal of the standard definition of the indicator function. A related concept...

Hilbert space gives an explicit example which is not a proper metric space. If h is a convex function, Lipschitz with constant 1 and h assumes its minimum...

Brouwer. It states that for any continuous function f {\displaystyle f} mapping a nonempty compact convex set to itself, there is a point x 0 {\displaystyle...

{\displaystyle \operatorname {sgn} x} there. Because the absolute value is a convex function, there is at least one subderivative at every point, including at the...

x_{n})=\mathrm {LSE} (0,x_{1},...,x_{n})} This function is a proper Bregman generator (strictly convex and differentiable). It is encountered in machine...

Moreover, the convex hull of the image of X under this embedding is dense in the space of probability measures on X. The delta function satisfies the...

Moreau-Yosida regularization) M f {\displaystyle M_{f}} of a proper lower semi-continuous convex function f {\displaystyle f} is a smoothed version of f {\displaystyle...

transformation on real-valued functions that are convex on a real variable. Specifically, if a real-valued multivariable function is convex on one of its independent...

(of arbitrary orientation) is (the interior of) a convex polytope, or in the 2-dimensional case a convex polygon. An open interval is a connected open set...

proximal operator is an operator associated with a proper, lower semi-continuous convex function f {\displaystyle f} from a Hilbert space X {\displaystyle...

Minkowski functional of any balanced set is a balanced function. Absorbing: If K {\textstyle K} is convex or balanced and if ( 0 , ∞ ) K = X {\textstyle (0...

are μ and ν. In convex analysis, the infimal convolution of proper (not identically + ∞ {\displaystyle +\infty } ) convex functions f 1 , … , f m {\displaystyle...