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An assumption of convex position can make certain computational problems easier to solve. For instance, the traveling salesman problem, NP-hard for arbitrary sets of points in the plane, is trivial for points in convex position: the optimal tour is the convex hull. [3]
In mathematics, a real-valued 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 between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set .
Convex analysis includes not only the study of convex subsets of Euclidean spaces but also the study of convex functions on abstract spaces. Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.
A convex curve (black) forms a connected subset of the boundary of a convex set (blue), and has a supporting line (red) through each of its points. A parabola, a convex curve that is the graph of the convex function () = In geometry, a convex curve is a plane curve that has a supporting line through each of its points.
Jensen's inequality generalizes the statement that a secant line of a convex function lies above its graph. Visualizing convexity and Jensen's inequality. In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function.
Convex function - a function in which the line segment between any two points on the graph of the function lies above the graph. Closed convex function - a convex function all of whose sublevel sets are closed sets. Proper convex function - a convex function whose effective domain is nonempty and it never attains minus infinity.
A convex set in light blue, and its extreme points in red. In mathematics , an extreme point of a convex set S {\displaystyle S} in a real or complex vector space is a point in S {\displaystyle S} that does not lie in any open line segment joining two points of S . {\displaystyle S.}
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, [1] whereas mathematical optimization is in general NP-hard. [2 ...