What is the relationship between convex functions and convex sets?

A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets.

What is affine set?

A set A is said to be an affine set if for any two distinct points, the line passing through these points lie in the set A. Note − S is an affine set if and only if it contains every affine combination of its points. Empty and singleton sets are both affine and convex set.

What does it mean if a set is convex?

A convex set is a set of points such that, given any two points A, B in that set, the line AB joining them lies entirely within that set. A convex set; no line can be drawn connecting two points that does not remain completely inside the set.

What is the difference between affine and convex?

An affine set contains every pairwise linear combination of points subject to the constraint that the co-efficients are real numbers. A convex set contains every pairwise linear combination of points subject to the condition that the co-efficients are real numbers that sum to 1.

What is the difference between affine and linear?

A linear function fixes the origin, whereas an affine function need not do so. An affine function is the composition of a linear function with a translation, so while the linear part fixes the origin, the translation can map it somewhere else.

What is concave and convex set?

Let f be a function of many variables, defined on a convex set S. We say that f is concave if the line segment joining any two points on the graph of f is never above the graph; f is convex if the line segment joining any two points on the graph is never below the graph.

What is the difference between a convex and an affine set?

A convex set contains every pairwise linear combination of points subject to the condition that the co-efficients are real numbers that sum to 1. So keeping it simple, an affine set is basically a convex set that happens to be infinite.

What is a convex set?

Convex set •A line segment defined by vectorsxandyis the set of points of the formαx + (1 − α)yforα ∈ [0,1] •A setC ⊂Rnis convex when, with any two vectorsxandythat belong to the setC, the line segment connectingxandyalso belongs toC Convex Optimization 8

Which set contains the line segment between two points?

A convex set contains the line segment between any two points in the set. Clearly if a set contains the line through a pair of points, then it also contains the line segment between the points.

What is an interior point of the set X?

A vectorx0is an interior point of the setX, if there is a ballB(x0,r) contained entirely in the setX Def. The interior of the setXis the set of all interior points ofX, denoted by