Mathematical function with convex lower level sets
For the unrelated generalization of convexity used in the calculus of variations, see Quasiconvexity (calculus of variations).
A quasiconvex function that is not convexA function that is not quasiconvex: the set of points in the domain of the function for which the function values are below the dashed red line is the union of the two red intervals, which is not a convex set.The probability density function of the normal distribution is quasiconcave but not concave.The bivariate normal joint density is quasiconcave.
In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form is a convex set. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. The negative of a quasiconvex function is said to be quasiconcave.
Quasiconvexity is a more general property than convexity in that all convex functions are also quasiconvex, but not all quasiconvex functions are convex. Univariate unimodal functions are quasiconvex or quasiconcave, however this is not necessarily the case for functions with multiple arguments. For example, the 2-dimensional Rosenbrock function is unimodal but not quasiconvex and functions with star-convex sublevel sets can be unimodal without being quasiconvex.
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In mathematics, a quasiconvexfunction is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse...
inequality Logarithmically convex function Pseudoconvex functionQuasiconvexfunction Subderivative of a convex function "Lecture Notes 2" (PDF). www.stat...
also true for a convex function, but it is not true for a quasiconvexfunction. Consider for example the quasiconvexfunction: f ( x ) = e x x 2 + 1 +...
E-invex functions were introduced by Abdulaleem as a generalization of differentiable convex functions. Convex function Pseudoconvex functionQuasiconvex function...
nonsingular Jacobian matrix. Quasiconvexfunctions and quasiconcave functions extend the concept of unimodality to functions whose arguments belong to higher-dimensional...
argument.[citation needed] Quasiconvexfunction Linear utility function - a special type of a quasilinear utility function. Varian, Hal (1992). Microeconomic...
lower-semicontinuity of the function implies that a function attains its minimum. The convexity of all the sublevel sets characterizes quasiconvexfunctions. Epigraph Level-set...
envelopes of subsets of the functions. For convex functions or quasiconvexfunctions, the upper envelope is again convex or quasiconvex. The lower envelope is...
optimization include the optimization of biconvex, pseudo-convex, and quasiconvexfunctions. Extensions of the theory of convex analysis and iterative methods...
function f such that ∇f · (y − x) ≥ 0 implies f(y) ≥ f(x) Quasiconvexfunction — function f such that f(tx + (1 − t)y) ≤ max(f(x), f(y)) for t ∈ [0,1]...
Infinitely divisible probability distributions Predictive inference Quasiconvexfunction "La prévision: ses lois logiques, ses sources subjectives", Annales...
refer to: Quasilinear function, a function that is both quasiconvex and quasiconcave Quasilinear utility, an economic utility function linear in one argument...
Analysis, Functions of Several Real Variables and Applications, Springer, January 2023, ISBN 978-3-031-04150-1. Approximation of quasiconvexfunctions, and...
Ellipsoid method: An iterative method for small problems with quasiconvex objective functions and of great theoretical interest, particularly in establishing...
{\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...
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...
Krzysztof C. (2001). "Convergence and efficiency of subgradient methods for quasiconvex minimization". Mathematical Programming, Series A. 90 (1). Berlin, Heidelberg:...
{\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...
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