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Concept in convex analysis
particular the subfields of convex analysis and optimization, a proper convex function is an extended real-valued convex function with a non-empty domain
Proper_convex_function
Mathematics of convex functions and sets
Convex analysis is the branch of mathematics that studies convex sets, convex functions, and their applications to optimization, functional analysis,
Convex_analysis
Terms in Maths
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
Closed_convex_function
Mathematical result in convex functions theory
a result in the theory of convex functions named after Werner Fenchel. Let f {\displaystyle f} be a proper convex function on R n {\displaystyle \mathbb
Fenchel's_duality_theorem
Cone of outward normals to a convex set at a point
+ ∞ } {\displaystyle f:V\to \mathbb {R} \cup \{+\infty \}} is a proper convex function, then its epigraph epi f = { ( x , t ) : t ≥ f ( x ) } {\displaystyle
Normal_cone_(convex_analysis)
Generalization of the Legendre transformation
optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known
Convex_conjugate
Theorem in convex analysis
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
Danskin's_theorem
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
Effective_domain
Smallest convex set containing a given set
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
Convex_hull
Property of functions which is weaker than continuity
in convex analysis. Given a convex (extended real) function, the epigraph might not be closed. But the lower semicontinuous hull of a convex function is
Semi-continuity
Mathematical set closed under positive linear combinations
nonnegative continuous functions is a convex cone. An affine convex cone is the set resulting from applying an affine transformation to a convex cone. A common
Convex_cone
Type of mathematical functions
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
Function of several complex variables
Function_of_several_complex_variables
Region above a graph
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
Epigraph_(mathematics)
Generalization of derivatives to real-valued functions
that point. Subderivatives arise in convex analysis, the study of convex functions, often in connection to convex optimization. Let f : I → R {\displaystyle
Subderivative
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
Glossary of Riemannian and metric geometry
Glossary_of_Riemannian_and_metric_geometry
Measure for evaluating probabilistic forecasts
and a convex class F {\displaystyle {\mathcal {F}}} of probability measures on ( Ω , A ) {\displaystyle (\Omega ,{\mathcal {A}})} . A function defined
Scoring_rule
American mathematician
Legendre–Fenchel transformation Proper convex function Subdifferential Subgradient Convex set Carathéodory's theorem Convex cone Duality (mathematics) Monotone
R._Tyrrell_Rockafellar
Theorem in topology
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
Brouwer_fixed-point_theorem
Mathematical optimization function
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
Moreau_envelope
Smooth approximation to the maximum function
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
LogSumExp
Function in mathematical optimization
proximal operator is an operator associated with a proper, lower semi-continuous convex function f {\displaystyle f} from a Hilbert space X {\displaystyle
Proximal_operator
Game where groups of players may enforce cooperative behaviour
are reversed, so that we say the cost game is convex if the characteristic function is submodular. Convex cooperative games have many nice properties:
Cooperative_game_theory
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
Busemann_function
exponential functions Inverse function Convex function, Concave function Singular function Harmonic function Weakly harmonic function Proper convex function Rational
List_of_real_analysis_topics
Concept in machine learning
H {\displaystyle H} indicates the Heaviside step function. However, this loss function is non-convex and non-smooth, and solving for the optimal solution
Loss functions for classification
Loss_functions_for_classification
graph. Closed convex function - a convex function all of whose sublevel sets are closed sets. Proper convex function - a convex function whose effective
List_of_convexity_topics
All numbers between two given numbers
asserts that the image of an interval by a continuous function is an interval; integrals of real functions are defined over an interval; etc. For example, interval
Interval_(mathematics)
Region underneath a graph
function is upper semicontinuous if and only if its hypograph is closed. Effective domain Epigraph (mathematics) – Region above a graph Proper convex
Hypograph_(mathematics)
Function whose values are sets (mathematics)
K.; Wąsowicz, S. (2013). "Hermite-Hadamard inequalities for convex set-valued functions". Demonstratio Mathematica. 46 (4): 655–662. doi:10.1515/dema-2013-0483
Set-valued_function
Function reducing distance between all points
closed under convex combinations, but not compositions. This class includes proximal mappings of proper, convex, lower-semicontinuous functions, hence it
Contraction_mapping
Mathematical transformation
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 real independent
Legendre_transformation
Function made from a set
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
Minkowski_functional
Mathematical function characterizing set membership
characteristic function in convex analysis, which is defined as if using the reciprocal of the standard definition of the indicator function. A related concept
Indicator_function
Concept in financial economics
distribution function g {\displaystyle g} if and only if g {\displaystyle g} is concave. If instead of the sublinear property,R is convex, then R is a
Coherent_risk_measure
Coherent measure for value at risk
measures, which are introduced in. Let g {\displaystyle g} be a convex proper function with g ( 1 ) = 0 {\displaystyle g(1)=0} and β {\displaystyle \beta
Entropic_value_at_risk
Mathematical theorem in convex analysis
of the following is true f {\displaystyle f} is a proper, lower semi-continuous, and convex function, f ≡ + ∞ {\displaystyle f\equiv +\infty } , or f ≡
Fenchel–Moreau_theorem
Convex and balanced set
of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of
Absolutely_convex_set
Theorem in real analysis
and is used to prove, the mean value theorem. If a real function f is continuous on a proper closed interval [a, b], differentiable on the open interval
Rolle's_theorem
locally convex spaces ( X , X ∗ ) {\displaystyle \left(X,X^{*}\right)} and ( Y , Y ∗ ) {\displaystyle \left(Y,Y^{*}\right)} . Then given the function f :
Perturbation_function
Loss function in machine learning
\mathbf {t} )\rangle )\end{aligned}}} . Hinge loss is a convex function, so many of the usual convex optimizers used in machine learning can work with it
Hinge_loss
Generalized function whose value is zero everywhere except at zero
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
Dirac_delta_function
Theorem on extension of bounded linear functionals
locally convex. However, suppose X is a topological vector space, not necessarily Hausdorff or locally convex, but with a nonempty, proper, convex, open
Hahn–Banach_theorem
Type of group used in topology and geometric group theory
generated group with a group action on a CAT(0) space that is geometrically proper, cocompact, and isometric. They form a possible notion of non-positively
CAT(0)_group
Objects that generalize functions
non-metrizable, locally convex topological vector space. The duality pairing between a distribution T in D′(U) and a test function φ {\displaystyle \varphi
Distribution (mathematical analysis)
Distribution_(mathematical_analysis)
n\geq 0} is log-convex. It also means that for every n {\displaystyle n} the function f ( n ) {\displaystyle f^{(n)}} is log-convex because ( log f
Absolutely and completely monotonic functions and sequences
Absolutely_and_completely_monotonic_functions_and_sequences
Class of convex shapes
the function on a union of sets equals the sum of its values on the sets. It is atom-free when every set whose function value is nonzero has a proper subset
Zonoid
Function returning minus 1, zero or plus 1
{\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
Sign_function
Constructions in nonsmooth analysis
\partial \delta _{C}=N_{C}^{\operatorname {conv} }.} For a proper, lower-semicontinuous, convex function f : H → R ¯ {\displaystyle f:H\to {\bar {\mathbb {R}
Normal cone (variational analysis)
Normal_cone_(variational_analysis)
Convex quadrilateral with at least one pair of parallel sides
usually considered to be a convex quadrilateral in Euclidean geometry, but there are also crossed cases. If shape ABCD is a convex trapezoid, then the ABDC
Trapezoid
Primal-Dual algorithm optimization for convex problems
designed to efficiently solve convex optimization problems that involve the minimization of a non-smooth cost function composed of a data fidelity term
Chambolle–Pock_algorithm
Vector space with a partial order
a proper cone if it is a convex cone satisfying C ∩ ( − C ) = { 0 } . {\displaystyle C\cap (-C)=\{0\}.} Explicitly, C {\displaystyle C} is a proper cone
Ordered_vector_space
Function that maps matrices to matrices
1]} . This definition is analogous to a concave scalar function. An operator convex function can be defined be switching ⪯ {\displaystyle \preceq } to
Analytic_function_of_a_matrix
Natural number
Problem for n = 4 {\displaystyle n=4} . There are 34 topologically distinct convex heptahedra, excluding mirror images. 34 is the magic constant of a 4 × 4
34_(number)
Topological vector spaces
{\displaystyle C_{\text{c}}^{\infty }(U)} into a complete Hausdorff locally convex TVS. The strong dual space of C c ∞ ( U ) {\displaystyle C_{\text{c}}^{\infty
Spaces of test functions and distributions
Spaces_of_test_functions_and_distributions
Any of 4 regular star polyhedra
polyhedra. They may be obtained by stellating and faceting the regular convex dodecahedron and icosahedron, and differ from these in having regular pentagrammic
Kepler–Poinsot_polyhedron
Set whose elements all belong to another set
It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called
Subset
Mathematical concept
three-space is an improper affine sphere. The graph of a locally strictly convex function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } is a
Affine_sphere
{\displaystyle f:X\to \mathbb {R} \cup \{+\infty \}} be a proper lower semicontinuous function that is bounded below (so inf f ( X ) ∈ R {\displaystyle
Ekeland's variational principle
Ekeland's_variational_principle
Algorithm for linear programming
x i ≥ 0 {\displaystyle \forall i,x_{i}\geq 0} is a (possibly unbounded) convex polytope. An extreme point or vertex of this polytope is known as basic
Simplex_algorithm
Technique to make a model more generalizable and transferable
convex, continuous, differentiable, with Lipschitz continuous gradient (such as the least squares loss function), and R {\displaystyle R} is convex,
Regularization_(mathematics)
Integral expressing the amount of overlap of one function as it is shifted over another
are μ and ν. In convex analysis, the infimal convolution of proper (not identically + ∞ {\displaystyle +\infty } ) convex functions f 1 , … , f m {\displaystyle
Convolution
Vector space with a notion of nearness
if it has a proper convex neighborhood of the origin. For any S ⊆ X {\displaystyle S\subseteq X} of a TVS X , {\displaystyle X,} the convex (resp. balanced
Topological_vector_space
convex shape in the plane that can cover any shape of diameter one Mahler's conjecture on the product of the volumes of a centrally symmetric convex body
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
In practice, the LJ heuristic has been recommended for functions that need be neither convex nor differentiable nor locally Lipschitz: The LJ heuristic
Luus–Jaakola
Type of feedforward neural network
non-linearly separable functions such as XOR. In the 1990s, MLPs competed directly with support vector machines, which offered convex optimization guarantees
Multilayer_perceptron
Infinitely detailed mathematical structure
mathematical treatment to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard Bolzano, Bernhard Riemann
Fractal
Geometric transformation that preserves lines but not angles nor the origin
be parallel after the transformation. convexity of sets: a convex set continues to be convex after the transformation. Moreover, the extreme points of
Affine_transformation
Experimental design that is optimal with respect to some statistical criterion
Bayesian experimental design Blocking (statistics) Computer experiment Convex function Convex minimization Design of experiments Efficiency (statistics) Entropy
Optimal_experimental_design
the function is convex. Well-known examples of convex functions include the quadratic function x 2 {\displaystyle x^{2}} and the exponential function e
Glossary_of_calculus
Isogonal polyhedron with regular faces
antiprisms, the convex polyhedrons as in 5 Platonic solids and 13 Archimedean solids—2 quasiregular and 11 semiregular—the non-convex star polyhedra as
Uniform_polyhedron
Class of algorithms for pattern analysis
linear adaptive filters and many others. Most kernel algorithms are based on convex optimization or eigenproblems and are statistically well-founded. Typically
Kernel_method
Term in mathematics
the following two conditions hold: X {\displaystyle X} is holomorphically convex, i.e. for every compact subset K ⊂ X {\displaystyle K\subset X} , the so-called
Stein_manifold
class is in one-to-one correspondence with a proper subset of itself (*124) relation A propositional function of some variables (usually two). This is similar
Glossary of Principia Mathematica
Glossary_of_Principia_Mathematica
locally convex topology – Space with topology generated by convex setsPages displaying short descriptions of redirect targets Sublinear function – Type
Discontinuous_linear_map
Type of vector space in math
variants, one simple statement is as follows: If f : H → R is a convex continuous function such that f(x) tends to +∞ when ‖x‖ tends to ∞, then f admits
Hilbert_space
Normed vector space that is complete
reflexive spaces to certain optimization problems. For example, every convex continuous function on the unit ball B {\displaystyle B} of a reflexive space attains
Banach_space
Overview of and topical guide to statistics
Metropolis–Hastings algorithm Importance sampling Mathematical optimization Convex optimization Linear programming Linear matrix inequality Quadratic programming
Outline_of_statistics
Measure of quantum entanglement in quantum mechanics
λ i {\displaystyle \lambda _{i}} are all of the eigenvalues. Is a convex function of ρ {\displaystyle \rho } : N ( ∑ i p i ρ i ) ≤ ∑ i p i N ( ρ i )
Negativity (quantum mechanics)
Negativity_(quantum_mechanics)
Concept in functional analysis
be dense in X {\displaystyle X} ). If Y {\displaystyle Y} is a locally convex space of dimension greater than 0 then this condition is also necessary
Positive_linear_operator
Topological invariant in mathematics
finitely additive, not-necessarily-nonnegative set function defined on finite unions of compact convex sets in ℝn that is "homogeneous of degree 0". For
Euler_characteristic
Average uncertainty in variable's states
\leq 1} . Accordingly, the negative entropy (negentropy) function is convex, and its convex conjugate is LogSumExp. The inspiration for adopting the word
Entropy_(information_theory)
valuation on convex sets and valuation on manifolds are a generalization of valuation in the sense of domain/measure theory. A valuation on convex sets is
Valuation_(measure_theory)
geodesic bicombing is convex. Every convex geodesic bicombing is conical, but the reverse implication does not hold in general. Every proper metric space with
Geodesic_bicombing
Set of vectors used to define coordinates
in a projective space of dimension n. A convex basis of a polytope is the set of the vertices of its convex hull. A cone basis consists of one point
Basis_(linear_algebra)
Family of iterative methods
{\displaystyle x} . The function M ( x ) {\displaystyle M(x)} has a unique point of maximum (minimum) and is strong concave (convex) The algorithm was first
Stochastic_approximation
optimization problems, such that the first problem in the sequence is convex (or nearly convex), the solution to each problem gives a good starting point to the
Graduated_optimization
Locally convex topological vector space
mathematics known as functional analysis, a reflexive space is a locally convex topological vector space for which the canonical evaluation map from X {\displaystyle
Reflexive_space
efficient point (proper minimizer) if x ¯ {\displaystyle {\bar {x}}} is a weakly efficient point with respect to a closed pointed convex cone C ~ {\displaystyle
Vector_optimization
manifold. Convex analysis the study of properties of convex functions and convex sets. Convex geometry part of geometry devoted to the study of convex sets
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Mathematical inequality in Sobolev space theory
with this issue with constant functions, for example, requiring trace zero, or subtracting the average over some proper subset of the domain. The constant
Poincaré_inequality
Second, the class of irrigation games is a non-convex cone which is a proper subset of the finite convex cone spanned by the duals of the unanimity games
Irrigation_game
Evolutionary algorithm
derivative-free methods for numerical optimization of non-linear or non-convex continuous optimization problems. They belong to the class of evolutionary
CMA-ES
In economics and consumer theory, a linear utility function is a function of the form: u ( x 1 , x 2 , … , x m ) = w 1 x 1 + w 2 x 2 + … w m x m {\displaystyle
Linear_utility
Organ of the lymphatic system
the proper functioning of the immune system, acting as filters for foreign particles including cancer cells, but have no detoxification function. In the
Lymph_node
Mathematical operator
the convex hull or affine hull of a subset of a vector space or the lower semicontinuous hull f ¯ {\displaystyle {\overline {f}}} of a function f : E
Closure_operator
Function of two vectors linear in each argument
use modules over a commutative ring R. It generalizes to n-ary functions, where the proper term is multilinear. For non-commutative rings R and S, a left
Bilinear_map
Eight bones that make up the wrist
columns. When considered as paired rows, each row forms an arch which is convex proximally and concave distally. On the palmar side, the carpus is concave
Carpal_bones
Bony process on the scapula (shoulder blade)
scapula. Its superior surface, directed upward, backward, and lateralward, is convex, rough, and gives attachment to some fibers of the deltoideus, and in the
Acromion
Five-pointed star polygon
five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle
Pentagram
Russian mathematician (born 1966)
in the field of convex geometry. His first published article studied the combinatorial structures arising from intersections of convex polyhedra.[P85]
Grigori_Perelman
travel, tourism, insurance
PROPER CONVEX-FUNCTION
PROPER CONVEX-FUNCTION
PROPER CONVEX-FUNCTION
PROPER CONVEX-FUNCTION
PROPER CONVEX-FUNCTION
PROPER CONVEX-FUNCTION
PROPER CONVEX-FUNCTION
PROPER CONVEX-FUNCTION
PROPER CONVEX-FUNCTION
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