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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
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
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
Function whose composition with the logarithm is convex
In mathematics, a function f {\displaystyle f} is logarithmically convex or superconvex if log ∘ f {\displaystyle {\log }\circ f} , the composition of
Logarithmically convex function
Logarithmically_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
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
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
Type of mathematical function
piecewise-differentiable functions. Important sub-classes of piecewise linear functions include the continuous piecewise linear functions and the convex piecewise linear
Piecewise_linear_function
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 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
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
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
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
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
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
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
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
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
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
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
Second-order deterministic global optimization algorithm
is a convex function. Let a function f ( x ) ∈ C 2 {\displaystyle {f({\boldsymbol {x}})\in C^{2}}} be a function of general non-linear non-convex structure
ΑΒΒ
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
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
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
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
{\displaystyle P_{i}(x_{i})=\cos(x_{i}/{\sqrt {i}})} . The non-linear and non-convex function is characterized by its unique multimodal structure, featuring multiple
Griewank_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)
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)
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
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)
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
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
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
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 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
Linear combination of points
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
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)
)\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
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
mirror Concave function, the negative of a convex function Concave polygon, a polygon which is not convex Concave set The concavity of a function, determined
Concave
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
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
Topics referred to by the same term
Uniform convexity can refer to: a uniformly convex space a uniformly convex function This disambiguation page lists articles associated with the title
Uniform_convexity
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
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
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)
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
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
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
Function returning minus 1, zero or plus 1
In mathematics, the sign function or signum function (from signum, Latin for "sign") is a function that has the value −1, +1 or 0 according to whether
Sign_function
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
Optimization performance test
optimization, the sphere function is a convex function used as a performance test problem for optimization algorithms. The sphere function was proposed by Kenneth
Sphere_function
and f : U → R m {\displaystyle f\colon U\to \mathbb {R} ^{m}} is a convex function, then f {\displaystyle f} has a second derivative almost everywhere
Alexandrov_theorem
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
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)
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
Mathematics concept
mathematics, and particularly convex analysis, the lower convex envelope f ˘ {\displaystyle {\breve {f}}} of a real-valued function f {\displaystyle f} defined
Lower_convex_envelope
Romanian mathematician
functional analysis and specializing in generalizations of the concept of a convex function. She was a winner of the Simion Stoilow Prize in mathematics. Elena
Elena_Moldovan_Popoviciu
Subfield of convex optimization
of convex optimization that studies problems consisting of minimizing a convex function over the intersection of an affine subspace and a convex cone
Conic_optimization
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
envelopes of subsets of the functions. For convex functions or quasiconvex functions, the upper envelope is again convex or quasiconvex. The lower envelope
Lower_envelope
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
Sums of sets of vectors are nearly convex
are sums of many functions. In probability, it can be used to prove a law of large numbers for random sets. A set is said to be convex if every line segment
Shapley–Folkman_lemma
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
terms of logarithmically convex functions, one in terms of increasing functions, and one in terms of non-negative functions. The conjecture has implications
Khabibullin's conjecture on integral inequalities
Khabibullin's_conjecture_on_integral_inequalities
Biconvex optimization is a generalization of convex optimization where the objective function and the constraint set can be biconvex. There are methods
Biconvex_optimization
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
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
Continuous function whose value increases to infinity
constrained optimization, a field of mathematics, a barrier function is a continuous function whose value increases to infinity as its argument approaches
Barrier_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
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
locally convex spaces ( X , X ∗ ) {\displaystyle \left(X,X^{*}\right)} and ( Y , Y ∗ ) {\displaystyle \left(Y,Y^{*}\right)} . Then given the function f :
Duality_gap
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
modulus of convexity is equivalent to a convex function in the following sense: there exists a convex function δ1(ε) such that δ ( ε / 2 ) ≤ δ 1 ( ε )
Modulus and characteristic of convexity
Modulus_and_characteristic_of_convexity
exponential functions Inverse function Convex function, Concave function Singular function Harmonic function Weakly harmonic function Proper convex function Rational
List_of_real_analysis_topics
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
Mathematics concept
the case of scalar functions of one variable the definition above is equivalent to usual monotonicity. Gradients of convex functions are cyclically monotone
Cyclical_monotonicity
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 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
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
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
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