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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
Generalisation of convexity
confused with the polysemetic concept of a quasiconvex function. A locally bounded Borel-measurable function f : R m × d → R {\textstyle f:\mathbb {R}
Quasiconvexity (calculus of variations)
Quasiconvexity_(calculus_of_variations)
Type of function
also true for a convex function, but it is not true for a quasiconvex function. Consider for example the quasiconvex function: f ( x ) = e x x 2 + 1 +
Pseudoconvex_function
Real function with secant line between points above the graph itself
inequality Logarithmically convex function Pseudoconvex function Quasiconvex function Subderivative of a convex function "Lecture Notes 2" (PDF). www.stat
Convex_function
Function in mathematical analysis
June 1992). Convex Functions, Partial Orderings, and Statistical Applications. Academic Press. p. 333. ISBN 9780080925226. Quasiconvex function v t e
Schur-convex_function
notions of convexity, quasiconvexity and rank-one convexity through the following diagram: f convex ⟹ f polyconvex ⟹ f quasiconvex ⟹ f rank-one convex
Polyconvex_function
envelopes of subsets of the functions. For convex functions or quasiconvex functions, the upper envelope is again convex or quasiconvex. The lower envelope is
Lower_envelope
Property of having a unique mode or maximum value
nonsingular Jacobian matrix. Quasiconvex functions and quasiconcave functions extend the concept of unimodality to functions whose arguments belong to higher-dimensional
Unimodality
Subset of a function's domain on which its value is equal
lower-semicontinuity of the function implies that a function attains its minimum. The convexity of all the sublevel sets characterizes quasiconvex functions. Epigraph Level-set
Level_set
the definition of type I functions introduced by Rueda and Hanson. Convex function Pseudoconvex function Quasiconvex function Hanson, Morgan A. (1981)
Invex_function
even more general statement: if f is a quasiconvex function of x for any fixed y, and a quasiconcave function of y for any fixed x, and the constraint
Min-max_optimization
Function linear in one argument, used in economics and consumer theory
argument.[citation needed] Quasiconvex function Linear utility function - a special type of a quasilinear utility function. Varian, Hal (1992). Microeconomic
Quasilinear_utility
Subfield of mathematical optimization
optimization include the optimization of biconvex, pseudo-convex, and quasiconvex functions. Extensions of the theory of convex analysis and iterative methods
Convex_optimization
Italian mathematician (1906–1985)
Theories of Probability, London: Routledge, 2000. Exchangeability Quasiconvex function "La prévision: ses lois logiques, ses sources subjectives", Annales
Bruno_de_Finetti
Let S = {f0, f1, ...} be a set of quasiconvex functions. Then the pointwise maximum maxi fi is itself quasiconvex, and the problem of finding the minimum
LP-type_problem
Italian mathematician
Analysis, Functions of Several Real Variables and Applications, Springer, January 2023, ISBN 978-3-031-04150-1. Approximation of quasiconvex functions, and
Paolo_Marcellini
Concept in mathematics
{\displaystyle Y} of a geodesic metric space X {\displaystyle X} is said to be quasiconvex if there is a constant C {\displaystyle C} such that any geodesic in
Hyperbolic_metric_space
Study of mathematical algorithms for optimization problems
Ellipsoid method: An iterative method for small problems with quasiconvex objective functions and of great theoretical interest, particularly in establishing
Mathematical_optimization
Peruvian mathematician
"Characterizing quasiconvexity of the pointwise infimum of a family of arbitrary translations of quasiconvex functions, with applications to sums and quasiconvex optimization"
Yboon_García_Ramos
function f such that ∇f · (y − x) ≥ 0 implies f(y) ≥ f(x) Quasiconvex function — function f such that f(tx + (1 − t)y) ≤ max(f(x), f(y)) for t ∈ [0,1]
List of numerical analysis topics
List_of_numerical_analysis_topics
American mathematician (1910–1985)
Tjalling C.; Debreu, Gérard (December 1982). "Additively decomposed quasiconvex functions" (PDF). Mathematical Programming. 24 (1). Springer: 1–38. doi:10
Tjalling_Koopmans
French economist and Nobel laureate (1921–2004)
Gérard; Koopmans, Tjalling C. (December 1982). "Additively decomposed quasiconvex functions" (PDF). Mathematical Programming. 24 (1): 1–38. doi:10.1007/BF01585092
Gérard_Debreu
Gives conditions that guarantee the max–min inequality holds with equality
Maurice Sion at 1958, is a further generalization, relaxing convexity to quasiconvexity. It states: Let X {\displaystyle X} be a convex subset of a linear topological
Minimax_theorem
Optimization method
lexmaxmin optimization[clarification needed] when the objectives are quasiconvex functions, and the feasible set X is a convex set. Yager presented a way to
Lexicographic max-min optimization
Lexicographic_max-min_optimization
Topics referred to by the same term
refer to: Quasilinear function, a function that is both quasiconvex and quasiconcave Quasilinear utility, an economic utility function linear in one argument
Quasilinear
Method for constructing existence proofs and calculating solutions in variational calculus
{\displaystyle x\in \Omega } , the function A ↦ F ( x , y , A ) {\displaystyle A\mapsto F(x,y,A)} is quasiconvex: there exists a cube D ⊆ R n {\displaystyle
Direct method in the calculus of variations
Direct_method_in_the_calculus_of_variations
Optimization algorithm
Krzysztof C. (2001). "Convergence and efficiency of subgradient methods for quasiconvex minimization". Mathematical Programming, Series A. 90 (1). Berlin, Heidelberg:
Stochastic_gradient_descent
Concept in mathematical optimization
Charnes–Cooper transformation. The objective function in a linear-fractional problem is both quasiconcave and quasiconvex (hence quasilinear) with a monotone property
Linear-fractional_programming
Mathematics of convex functions and sets
the calculus of variations, rank-one convexity, polyconvexity, and quasiconvexity arise in vector-valued variational problems. In metric geometry and
Convex_analysis
Approximation method in statistics
initially decrease rapidly, it can converge to a nonstationary point on quasiconvex problems, by an example of M. J. D. Powell. More detailed descriptions
Non-linear_least_squares
Prediction of digital video quality
is a distance function. The square of such a function is not convex, but is locally convex and quasiconvex, making SSIM a feasible target for optimization
Structural similarity index measure
Structural_similarity_index_measure
Smallest convex set containing a given set
convex hull of the weight vectors of solutions. One can maximize any quasiconvex combination of weights by finding and checking each convex hull vertex
Convex_hull
American mathematician
Indiana Univ. Math. J. 33 (1984), no. 5, 773–797. Evans, Lawrence C. Quasiconvexity and partial regularity in the calculus of variations. Arch. Rational
Lawrence_C._Evans
French mathematician (born 1944)
problem. Their study of duality gaps was extended by Di Guglielmo to the quasiconvex closure of a non-convex minimization problem—that is, the problem defined
Ivar_Ekeland
Sums of sets of vectors are nearly convex
problem. Their study of duality gaps was extended by Di Guglielmo to the quasiconvex closure of a non-convex minimization problem—that is, the problem defined
Shapley–Folkman_lemma
Graduate-level textbooks in mathematics
Zhang 2020-11-03 218 9780691202525 209 The Structure of Groups with a Quasiconvex Hierarchy Daniel T. Wise 2021-05-04 376 9780691170459 210 Global Nonlinear
Annals_of_Mathematics_Studies
Mathematical space
(2009-10-29), "Research announcement: The structure of groups with a quasiconvex hierarchy", Electronic Research Announcements in Mathematical Sciences
3-manifold
\Lambda _{\partial G}(H)} . If H ≤ G is quasi-isometrically embedded (i.e. quasiconvex) subgroup, then the Cannon–Thurston map ∂i: ∂H → ∂G exists and is a topological
Cannon–Thurston_map
invective, inveigh, nonconvective, pretervection, provection, quasiconvex, quasiconvexity, transvection, vection, vector, vectorial, vecture, vehicle,
List of Latin verbs with English derivatives
List_of_Latin_verbs_with_English_derivatives
American mathematician
(2012). "The quasiconvex envelope through first-order partial differential equations which characterize quasiconvexity of nonsmooth functions". Discrete
Robert_R._Jensen
American mathematician (1927-1984)
characterization of lower semicontinuous variational problems in terms of quasiconvexity. He greatly contributed to the solution of Hilbert's nineteenth and
Charles_B._Morrey_Jr.
QUASICONVEX FUNCTION
QUASICONVEX FUNCTION
Male
Celtic
, great justiciary, or functionary.
Surname or Lastname
English
English : nickname from the animal, Middle English catte ‘cat’. The word is found in similar forms in most European languages from very early times (e.g. Gaelic cath, Slavic kotu). Domestic cats were unknown in Europe in classical times, when weasels fulfilled many of their functions, for example in hunting rodents. They seem to have come from Egypt, where they were regarded as sacred animals.English : from a medieval female personal name, a short form of Catherine.Variant spelling of German and Dutch Katt.
Male
Egyptian
, a great functionary.
Surname or Lastname
English
English : topographic name for someone who lived by the gates of a medieval walled town. The Middle English singular gate is from the Old English plural, gatu, of geat ‘gate’ (see Yates). Since medieval gates were normally arranged in pairs, fastened in the center, the Old English plural came to function as a singular, and a new Middle English plural ending in -s was formed. In some cases the name may refer specifically to the Sussex place Eastergate (i.e. ‘eastern gate’), known also as Gates in the 13th and 14th centuries, when surnames were being acquired.Americanized spelling of German Götz (see Goetz).Translated form of French Barrière (see Barriere).In New England, Gates was the preferred English version of the name of an extensive French family, called Barrière dit Langevin.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Surname or Lastname
English
English : occupational name for a dresser of cloth, Old English fullere (from Latin fullo, with the addition of the English agent suffix). The Middle English successor of this word had also been reinforced by Old French fouleor, foleur, of similar origin. The work of the fuller was to scour and thicken the raw cloth by beating and trampling it in water. This surname is found mostly in southeast England and East Anglia. See also Tucker and Walker.In a few cases the name may be of German origin with the same form and meaning as 1 (from Latin fullare).Americanized version of French Fournier.Samuel Fuller (1589–1633), born in Redenhall, Norfolk, England, was among the Pilgrim Fathers who sailed on the Mayflower in 1620. He was a deacon of the church and until his death functioned as Plymouth Colony’s physician.
Biblical
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Male
Egyptian
, a high Egyptian functionary.
Surname or Lastname
English (chiefly Kent and Sussex)
English (chiefly Kent and Sussex) : occupational name for a designer or engineer, from a Middle English reduced form of Old French engineor ‘contriver’ (a derivative of engaigne ‘cunning’, ‘ingenuity’, ‘stratagem’, ‘device’). Engineers in the Middle Ages were primarily designers and builders of military machines, although in peacetime they might turn their hands to architecture and other more pacific functions.German : from the Latin personal name Januarius (see January 1). Jänner is a South German word for ‘January’, and so it is possible that this is one of the surnames acquired from words denoting months of the year, for example by converts who had been baptized in that month, people who were born or baptized in that month, or people whose taxes were due in January.
Male
Egyptian
, an Egyptian functionary.
Male
Egyptian
, Functionary of the Interior.
Male
Egyptian
, an Egyptian functionary.
Male
Egyptian
, the son of the functionary Heknofre.
QUASICONVEX FUNCTION
QUASICONVEX FUNCTION
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QUASICONVEX FUNCTION
QUASICONVEX FUNCTION
QUASICONVEX FUNCTION
QUASICONVEX FUNCTION