What does it mean if a function is strictly convex?
A strictly convex function is a function that the straight line between any pair of points on the curve is above the curve. except for the intersection points between the straight line and the curve.
How do you determine strict convexity?
We can determine the concavity/convexity of a function by determining whether the Hessian is negative or positive semidefinite, as follows. if H(x) is positive definite for all x ∈ S then f is strictly convex.
What is strong convexity?
Intuitively speaking, strong convexity means that there exists a quadratic lower bound on the growth of the function. This directly implies that a strong convex function is strictly convex since the quadratic lower bound growth is of course strictly grater than the linear growth.
What does convexity mean in economics?
Convexity is a measure of the curvature in the relationship between bond prices and bond yields. Convexity demonstrates how the duration of a bond changes as the interest rate changes.
What is the meaning of convexity?
: the quality or state of being curved outward : the quality or state of being convex. : a shape that is curved outward : a convex shape. See the full definition for convexity in the English Language Learners Dictionary.
How do you know if a function is convex?
For a twice-differentiable function f, if the second derivative, f ”(x), is positive (or, if the acceleration is positive), then the graph is convex (or concave upward); if the second derivative is negative, then the graph is concave (or concave downward).
What is convex function in machine learning?
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets. Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard.
Does strict convexity imply strong convexity?
These conditions are given in increasing order of strength; strong convexity implies strict convexity which implies convexity. Geometrically, convexity means that the line segment between two points on the graph of f lies on or above the graph itself.
What is strict convexity of indifference curve?
So, in two dimensions, with strictly monotonic preferences, strict convexity says that if two consumption bundles are each on the same indifference curve as x, then any point on a line connecting these two points (except for the points themselves) will be on a higher indifference curve than x.
What does it mean when a function is strictly convex?
Strict convexity means that the line segment lies strictly above the graph off, except at the segmentendpoints. (So actually the function in the gure appears to be strictly convex.) 3.1 Consequences of convexity Why do we care if a function is (strictly/strongly) convex?
Which is the best definition of strong convexity?
Strong convexity is one formulation that allows us to talk about how “convex” or “curved” a convex function is. is strongly convex with parameter if Equation is just like Equation except the RHS has an added negative term which makes it smaller. If is differentiable, being strongly convex with parameter is equivalent to
Can a strictly convex line have a slope?
In order for a line to be convex (or express convexity) there has to be a slope to the line. For those that have taken calculus, a strictly convex line has to have a second derivative that is greater than zero. Graphically, this means that a straight line cannot be strictly convex,…
Do you need convexity to have an indifference curve?
Strict convexity isn’t needed to have an indifference curve, but without it, we are assuming that the two goods are perfect substitutes, which isn’t likely. Additionally, tangency can only be achieved when preferences are well-behaved/strictly convex. This is because of the linear nature of a budget constraint.