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Real function with secant line between points above the graph itself
function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph of the function
Convex_function
Function whose composition with the logarithm is convex
In mathematics, a function f is logarithmically convex or superconvex if log ∘ f {\displaystyle {\log }\circ f} , the composition of the logarithm with
Logarithmically convex function
Logarithmically_convex_function
In geometry, set whose intersection with every line is a single line segment
the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets
Convex_set
Function in mathematical analysis
In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function f : R d → R {\displaystyle
Schur-convex_function
Mathematical function with convex lower level sets
In mathematics, a quasiconvex function is a real-valued function defined on a convex subset of a real vector space, such that for any real number y, the
Quasiconvex_function
Negative of a convex function
concave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to that convex combination
Concave_function
Type of function in linear algebra
and positive homogeneity implies the third. Every sublinear function is a convex function: For 0 ≤ t ≤ 1 , {\displaystyle 0\leq t\leq 1,} p ( t x + (
Sublinear_function
Subfield of mathematical optimization
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently
Convex_optimization
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
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
Type of mathematical function
piecewise-differentiable functions, PDIFF. Important sub-classes of piecewise linear functions include the continuous piecewise linear functions and the convex piecewise
Piecewise_linear_function
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
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 of convex functions
mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building
Jensen's_inequality
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
Type of plane curve
Examples of convex curves include the convex polygons, the boundaries of convex sets, and the graphs of convex functions. Important subclasses of convex curves
Convex_curve
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
In the field of mathematics known as convex analysis, the indicator function of a set is a convex function that indicates the membership (or non-membership)
Indicator function (convex analysis)
Indicator_function_(convex_analysis)
Distance from origin of tangent hyperplanes
In mathematics, the support function hA of a non-empty closed convex set A in R n {\displaystyle \mathbb {R} ^{n}} describes the (signed) distances of
Support_function
Study of mathematical algorithms for optimization problems
Generally, unless the objective function is convex in a minimization problem, there may be several local minima. In a convex problem, if there is a local
Mathematical_optimization
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
Type of mathematical function
In convex analysis, a non-negative function f : Rn → R+ is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it
Logarithmically concave function
Logarithmically_concave_function
Mathematical function
K-convex functions, first introduced by Scarf, are a special weakening of the concept of convex function which is crucial in the proof of the optimality
K-convex_function
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 independent
Legendre_transformation
Class of mathematical functions
Intuitively, subharmonic functions are related to convex functions of one variable as follows. If the graph of a convex function and a line intersect at
Subharmonic_function
Minimal superset that intersects each axis-parallel line in an interval
orthogonal convex hull is not defined using properties of sets, but properties of functions about sets. Namely, it restricts the notion of convex function as
Orthogonal_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
Theorem in optimal transport
an absolutely continuous probability measure is the gradient of a convex function. More precisely, if μ {\displaystyle \mu } and ν {\displaystyle \nu
Brenier's_theorem
Principle in mathematical optimization
with replacing a non-convex function with its convex closure, that is the function that has the epigraph that is the closed convex hull of the original
Duality_(optimization)
Theorem on extension of bounded linear functionals
1.} Every sublinear function is a convex function. On the other hand, if p : X → R {\displaystyle p:X\to \mathbb {R} } is convex with p ( 0 ) ≥ 0 , {\displaystyle
Hahn–Banach_theorem
Concept in economics
relation is convex, but not strictly-convex. 3. A preference relation represented by linear utility functions is convex, but not strictly convex. Whenever
Convex_preferences
Gives conditions that guarantee the max–min inequality holds with equality
compact and convex, and to functions that are concave in their first argument and convex in their second argument (known as concave-convex functions). Formally
Minimax_theorem
Type of function
In convex analysis and the calculus of variations, both branches of mathematics, a pseudoconvex function is a function that behaves like a convex function
Pseudoconvex_function
Topics referred to by the same term
Convex function, when the line segment between any two points on the graph of the function lies above or on the graph Convex conjugate, of a function
Convex
Function used as a performance test problem for optimization algorithms
In mathematical optimization, the Rosenbrock function is a non-convex function, introduced by Howard H. Rosenbrock in 1960, which is used as a performance
Rosenbrock_function
Smooth approximation to the maximum function
this formula internally. LSE is convex but not strictly convex. We can define a strictly convex log-sum-exp type function by adding an extra argument set
LogSumExp
Measure of difference between two points
measure of difference between two points, defined in terms of a strictly convex function; they form an important class of divergences. When the points are interpreted
Bregman_divergence
Extension of the factorial function
is the unique interpolating function for the factorial, defined over the positive reals, which is logarithmically convex, meaning that y = log f ( x
Gamma_function
Topics referred to by the same term
Strictly convex may refer to: Strictly convex function, a function having the line between any two points above its graph Strictly convex polygon, a polygon
Strictly_convex
Order-preserving mathematical function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept
Monotonic_function
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)
Function used as a performance test problem for optimization algorithms
Rastrigin function of two variables In mathematical optimization, the Rastrigin function is a non-convex function used as a performance test problem for
Rastrigin_function
related to the problems on convex sets is the following problem on a convex function f: Rn → R: Strong unconstrained convex function minimization (SUCFM):
Algorithmic problems on convex sets
Algorithmic_problems_on_convex_sets
Linear combination of points where all coefficients are non-negative and sum to 1
In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points
Convex_combination
Representation of a mathematical function
y)=-(\cos(x^{2})+\cos(y^{2}))^{2}.} Asymptote Chart Plot Concave function Convex function Contour plot Critical point Derivative Epigraph Normal to a graph
Graph_of_a_function
self-concordant barrier is a particular self-concordant function, that is also a barrier function for a particular convex set. Self-concordant barriers are important
Self-concordant_function
Algebra theorem about convex functions
majorization inequality, is a theorem in elementary algebra for convex and concave real-valued functions, defined on an interval of the real line. It generalizes
Karamata's_inequality
Matrix of second derivatives
Hessian determinant is a polynomial of degree 3. The Hessian matrix of a convex function is positive semi-definite. Refining this property allows us to test
Hessian_matrix
Algorithms for solving convex optimization problems
a convex function and G is a convex set. Without loss of generality, we can assume that the objective f is a linear function. Usually, the convex set
Interior-point_method
Mathematical optimization function
regularization) M f {\displaystyle M_{f}} of a proper lower semi-continuous convex function f {\displaystyle f} is a smoothed version of f {\displaystyle f} .
Moreau_envelope
Function used as a performance test problem for optimization algorithms
In mathematical optimization, the Ackley function is a non-convex function used as a performance test problem for optimization algorithms. It was proposed
Ackley_function
f} is convex. Every convex function is polyconvex. For the case m = n {\displaystyle m=n} , the determinant function is polyconvex, but not convex. In particular
Polyconvex_function
Function in mathematical optimization
operator is an operator associated with a proper, lower semi-continuous convex function f {\displaystyle f} from a Hilbert space X {\displaystyle {\mathcal
Proximal_operator
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
non-concave function to a concave function. A related concept is convexification – converting a non-convex function to a convex function. It is especially
Concavification
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
Concept in mathematics of vector spaces
uniformly convex. Conversely, L ∞ {\displaystyle L^{\infty }} is not uniformly convex. Modulus and characteristic of convexity Uniformly convex function Uniformly
Uniformly_convex_space
Topics referred to by the same term
In mathematics, a convex graph may be a convex bipartite graph a convex plane graph the graph of a convex function This disambiguation page lists articles
Convex_graph
Concept in mathematical optimization
variable chosen from a convex subset of R n {\displaystyle \mathbb {R} ^{n}} , f {\displaystyle f} is the objective or utility function, g i ( i = 1 , …
Karush–Kuhn–Tucker_conditions
Study of optimal transportation and allocation of resources
are both optimal. If, on the other hand, we choose the strictly convex cost function proportional to the square of Euclidean distance ( c ( x , y ) =
Transportation theory (mathematics)
Transportation_theory_(mathematics)
Middle quantile of a data set or probability distribution
single point or an empty set). Every convex function is a C function, but the reverse does not hold. If f is a C function, then f ( med [ X ] ) ≤ med [
Median
Length in a vector space
seminorm is a sublinear function and thus satisfies all properties of the latter. In particular, every norm is a convex function. The concept of unit circle
Norm_(mathematics)
Mathematical inequality about convex functions
In convex analysis, Popoviciu's inequality is an inequality about convex functions. It is similar to Jensen's inequality and was found in 1965 by Tiberiu
Popoviciu's_inequality
Distance function defined between probability distributions
particularly simple way to state that a function is c-convex in this case: a function f {\displaystyle f} is c-convex iff it is Lipschitz, with Lipschitz
Wasserstein_metric
Preorder on vectors of real numbers
x j ) {\displaystyle \varepsilon \in (0,x_{i}-x_{j})} . For every convex function h : R → R {\displaystyle h:\mathbb {R} \to \mathbb {R} } , ∑ i = 1
Majorization
Type of function in complex analysis
is an analytic function on an open set, then log | f | {\displaystyle \log |f|} is plurisubharmonic on that open set. Convex functions are plurisubharmonic
Plurisubharmonic_function
Iterative method for minimizing convex functions
the ellipsoid method is an iterative method for minimizing convex functions over convex sets. The ellipsoid method generates a sequence of ellipsoids
Ellipsoid_method
Method to solve optimization problems
of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set
Linear_programming
Concept in convex optimization mathematics
: R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } be a convex function with domain R n . {\displaystyle \mathbb {R} ^{n}.} A classical subgradient
Subgradient_method
Concept in mathematics
optimization over particular geometries. We are given convex function f {\displaystyle f} to optimize over a convex set K ⊂ R n {\displaystyle K\subset \mathbb
Mirror_descent
Form of projection
^{d}\rightarrow \mathbb {R} ,\ i=1,\dots ,n} are possibly non-differentiable convex functions. The lack of differentiability rules out conventional smooth optimization
Proximal_gradient_method
Invex functions were introduced by Hanson as a generalization of convex functions. Ben-Israel and Mond provided a simple proof that a function is invex
Invex_function
F_{t}} is convex. Since the Busemann function B γ {\displaystyle B_{\gamma }} is the pointwise limit of F t {\displaystyle F_{t}} , Busemann functions are convex
Busemann_function
Method for finding stationary points of a function
the second derivative is positive, the quadratic approximation is a convex function of t {\displaystyle t} , and its minimum can be found by setting the
Newton's method in optimization
Newton's_method_in_optimization
Solution process for some optimization problems
objective function is concave (maximization problem), or convex (minimization problem) and the constraint set is convex, then the program is called convex and
Nonlinear_programming
)\leq 0} , where f is a bounded function and g is a convex function. MMO problems play a central role in game theory, convex optimization and online machine
Min-max_optimization
Product of numbers from 1 to n
Bohr–Mollerup theorem, which states that the gamma function (offset by one) is the only log-convex function on the positive real numbers that interpolates
Factorial
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
Set-to-real map with diminishing returns
\sum _{S}\alpha _{S}=1,\alpha _{S}\geq 0\right)} . The convex closure of any set function is convex over [ 0 , 1 ] n {\displaystyle [0,1]^{n}} . Consider
Submodular_set_function
Statistical distribution for dependence between random variables
\rightarrow [0,\infty )\ } is a continuous, strictly decreasing and convex function such that ψ ( 1 ; θ ) = 0 , {\displaystyle \ \psi (1;\theta )=0\
Copula_(statistics)
Mathematical Theory
an uncountably infinite (and possibly non-convex) collection of real-valued vectors, and so on. The functions P(), Y_i() are also arbitrary and do not
Drift_plus_penalty
Length of a line segment
strictly convex function of the two points, unlike the distance, which is non-smooth (near pairs of equal points) and convex but not strictly convex. The
Euclidean_distance
Mathematics concept
In mathematics, the lower convex envelope f ˘ {\displaystyle {\breve {f}}} of a function f {\displaystyle f} defined on an interval [ a , b ] {\displaystyle
Lower_convex_envelope
Concept in Hlibert spaces mathematics
in fact, not operator monotone! A function f : I → R {\displaystyle f:I\to \mathbb {R} } is said to be operator convex if for all n {\displaystyle n} and
Trace_inequality
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
Concept in mathematical finance
in probability theory: the expected value of a convex function is greater than or equal to the function of the expected value: E [ f ( X ) ] ≥ f ( E [
Convexity_(finance)
American mathematician
2013.03.001. Convex analysis (cf. Werner Fenchel) Convex function Characteristic function (convex analysis) Closed convex function Convex conjugate Epigraph
R._Tyrrell_Rockafellar
Mathematical concept
} This follows immediately from the definition of the convex conjugate. For a convex function f {\displaystyle f} this also follows from the Legendre
Young's inequality for products
Young's_inequality_for_products
Cone of outward normals to a convex set at a point
In convex analysis and optimization, the normal cone to a set at a point is a convex cone consisting of vectors that make a non-acute angle with every
Normal_cone_(convex_analysis)
operator convex functions, and are encountered in operator theory and in matrix theory, and led to the Löwner–Heinz inequality. Operator monotone functions are
Operator_monotone_function
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
Mathematical set closed under positive linear combinations
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
Convex_cone
Right continuous function with left limits
}} of any convex function f {\displaystyle f} defined on an open interval, is an increasing cadlag function. The set of all càdlàg functions from E {\displaystyle
Càdlàg
Rounding of an interior or exterior corner
interior corner is a line of concave function, whereas a fillet on an exterior corner is a line of convex function (in these cases, fillets are typically
Fillet_(mechanics)
Property of a mathematical matrix
a point p , {\displaystyle p,} then the function is convex near p, and, conversely, if the function is convex near p , {\displaystyle p,} then the Hessian
Definite_matrix
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
Triangle inequality in Lp spaces
{\textstyle \phi (x)} is a convex function of x . {\textstyle x.} log ϕ ( x ) {\textstyle \log \phi (x)} is a convex function of log ( x ) . {\textstyle
Minkowski_inequality
Scalar physical quantities representing system states
)}_{T,N}\geq 0} Where Helmholtz energy is a concave function of temperature and convex function of volume. ( ∂ 2 H ∂ P 2 ) S , N ≤ 0 {\displaystyle {\biggl
Thermodynamic_potential
Average value of a random variable
applications to probability theory. Jensen's inequality: Let f: R → R be a convex function and X a random variable with finite expectation. Then f ( E ( X )
Expected_value
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
CONVEX FUNCTION
CONVEX FUNCTION
Boy/Male
Irish American
Hound lover. Full of desire; much desire.
Surname or Lastname
Italian
Italian : from the title of rank conte ‘count’ (from Latin comes, genitive comitis ‘companion’). Probably in this sense (and the Late Latin sense of ‘traveling companion’), it was a medieval personal name; as a title it was no doubt applied ironically as a nickname for someone with airs and graces or simply for someone who worked in the service of a count.English : variant of Count, cognate with 1.French : nickname for someone in the service of a count or for someone who behaved pretentiously, from Old French conte, cunte ‘count’ (of the same derivation as 1).French (Conté) : variant of Comté (see Comte).
Boy/Male
American, Christian, German, Indian
High Desire
Boy/Male
Indian, Kannada, Tamil
God Murugan
Boy/Male
Irish American
Strong willed or wise. Also a : Hero.
Boy/Male
British, Christian, English
Wagoner; To Convey
Boy/Male
Irish
Hound of the plains.
Surname or Lastname
English
English : metathesized form of the occupational name Coyner.English : possibly an occupational name for a dealer in rabbits or rabbit skins, from an agent derivative of Middle English cony ‘rabbit’ (see Coney).
Male
English
Variant spelling of English Connor, CONNER means "hound-lover."
Male
English
Anglicized form of Irish Gaelic Conláed, CONLEY means "purifying fire."
Boy/Male
American, British, English
Shepherd
Surname or Lastname
English
English : habitational name from a place named Cove, examples of which are found in Devon, Hampshire, and Suffolk, from Old English cofa ‘cove’, ‘bay’, ‘inlet’, also ‘shelter’, ‘hut’, or a topographic name with the same meaning.
Surname or Lastname
Irish
Irish : variant spelling of Connor, now common in Scotland.English : occupational name for an inspector of weights and measures, Middle English connere, cunnere ‘inspector’, an agent derivative of cun(nen) ‘to examine’.
Surname or Lastname
Spanish and Portuguese
Spanish and Portuguese : nickname from the title of rank conde ‘count’, a derivative of Latin comes, comitis ‘companion’.English : unexplained.
Surname or Lastname
English
English : from Old French covine ‘fraud’, ‘deceit’, hence a derogatory nickname for a trickster.English : habitational name from a place in Staffordshire named Coven ‘(place) at the huts or shelters (Old English cofa, dative plural cofum)’.
Surname or Lastname
English
English : unexplained.
Boy/Male
Irish
Hero.
Surname or Lastname
English (Leicestershire)
English (Leicestershire) : variant of Culver.
Boy/Male
American, British, English
Dove
Surname or Lastname
English
English : from Middle English cony ‘rabbit’ (a back-formation from conies, from Old French conis, plural of conil), a nickname for someone thought to resemble a rabbit in some way or a metonymic occupational name for a dealer in rabbits or rabbit skins.
CONVEX FUNCTION
CONVEX FUNCTION
Female
English
Variant spelling of English Lakeisha, LAKISHA means "cassia," a bark similar to cinnamon.
Boy/Male
Tamil
Ramchandra | ராமசஂதà¯à®°à®¾Â
Lord Rama
Boy/Male
Hindu
Auspicious, Lord Shiva
Female
Norse
 Variant spelling of Old Norse Auðr, AUÃA means "deeply rich."
Boy/Male
Tamil
Delightful
Boy/Male
British, English
Lives Near the Stag's Spring
Surname or Lastname
English
English : presumably a patronymic from a Middle English survival of Old English Ramm ‘ram’ or Hrafn ‘raven’ as a personal name.Name found among people of Indian origin in Guyana and Trinidad : probably from the personal name Ram and the English suffix -son.
Boy/Male
Indian, Punjabi, Sikh
Brave and Happy
Surname or Lastname
English (Devon and Cornwall)
English (Devon and Cornwall) : probably a variant spelling of Guest.
Girl/Female
Hindu
Blossomed, Flowers in bloom
CONVEX FUNCTION
CONVEX FUNCTION
CONVEX FUNCTION
CONVEX FUNCTION
CONVEX FUNCTION
a.
Convex on both sides; double convex. See under Convex, a.
v. t.
To call before a judge or judicature; to summon; to convene.
n.
The conger eel; -- called also congeree.
v. t.
To cause to pass from one place or person to another; to serve as a medium in carrying (anything) from one place or person to another; to transmit; as, air conveys sound; words convey ideas.
a.
Concave on one side and convex on the other, as an eggshell or a crescent.
v. t.
To context.
n.
A convex body or surface.
v. t.
To exchange for some specified equivalent; as, to convert goods into money.
a.
Convex on one side, and flat on the other; plano-convex.
a.
Convex on one side, and concave on the other. The curves of the convex and concave sides may be alike or may be different. See Meniscus.
a.
Convex on both sides; as, a biconvex lens.
adv.
In a convex form; as, a body convexly shaped.
imp. & p. p.
of Cove
v. t.
To impart or communicate; as, to convey an impression; to convey information.
a.
Made convex; protuberant in a spherical form.
n. & v.
See Conge, Conge.
a.
Specifically, having such a combination of concave and convex sides as makes the focal axis the shortest line between them. See Illust. under Lens.
v. t.
To accompany; to convoy.
a.
Plane or flat on one side, and convex on the other; as, a plano-convex lens. See Convex, and Lens.
dv.
In a convex form; convexly.