What is the relation between convexity and quasi convexity?

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. All convex functions are also quasiconvex, but not all quasiconvex functions are convex, so quasiconvexity is a generalization of convexity.

Is quasi concave the same as convex?

The notion of quasiconcavity is weaker than the notion of concavity, in the sense that every concave function is quasiconcave. Similarly, every convex function is quasiconvex. A concave function is quasiconcave. A convex function is quasiconvex.

Can discontinuous functions be convex?

Thus, a discontinuous convex function is unbounded on any interior interval and is not measurable. If, for some function f, inequality (2) is true for any two points x1 and x2 in some interval and any p1>0 and p2>0, the function f is continuous and, of course, convex on this interval.

Can a function be convex and concave at the same time?

An affine function (f (x) = ax + b) is simultaneously convex and concave. A differentiable function f is concave on an interval if its derivative function f ′ is decreasing on that interval: a concave function has a decreasing slope.

How do you prove a function is quasi concave?

The function f is strictly quasi-concave iff for any x, x ∈ C, if x = x and f(x) ≥ f(x) then for any θ ∈ (0,1), setting xθ = θx + (1 − θ)x, f(xθ) > f(x). The function f is quasi-convex iff −f is quasi-concave. It is strictly quasi-convex iff −f is strictly quasi-concave.

How do you determine if a function is quasi concave?

Reminder: A function f is quasiconcave if and only if for every x and y and every λ with 0 ≤ λ ≤ 1, if f(x) ≥ f(y) then f((1 − λ)x + λy) ≥ f(y). Suppose that the function U is quasiconcave and the function g is increasing.

How do you know if a function is quasi concave?

Reminder: A function f is quasiconcave if and only if for every x and y and every λ with 0 ≤ λ ≤ 1, if f(x) ≥ f(y) then f((1 − λ)x + λy) ≥ f(y). Suppose that the function U is quasiconcave and the function g is increasing. Show that the function f defined by f(x) = g(U(x)) is quasiconcave. Suppose that f(x) ≥ f(y).

Are convex functions continuous?

A convex function is a continuous function whose value at the midpoint of every interval in its domain does not exceed the arithmetic mean of its values at the ends of the interval.

Can a discontinuous function be concave or convex?

Can a convex function be discontinuous? – Quora. Yes, on the boundary of its domain. Consider: and that is because convexity of a function is a property of its epigraph (the points above the graph of the function).

Can a function be neither convex or concave?

Note that it is possible for f to be neither convex nor concave. We say that the convexity/concavity is strict if the graph of f(x) over the interval I contains no straight line segments.

How do you determine if a function is convex or concave Hessian?

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 an example of a quasiconvex function that is not convex?

Any monotonic function is both quasiconvex and quasiconcave. More generally, a function which decreases up to a point and increases from that point on is quasiconvex (compare unimodality ). is an example of a quasiconvex function that is neither convex nor continuous.

How do you find quasiconvexity?

Quasiconvexity A function f : X → IR∪{+∞} is said to be quasiconvex on K if, for all λ ∈ IR, the sublevel set Sλ= {x ∈ X : f(x) ≤ λ} is convex. – p.3/48 Quasiconvexity A function f : X → IR∪{+∞} is said to be quasiconvex on K if, for all λ ∈ IR, the sublevel set Sλ= {x ∈ X : f(x) ≤ λ} is convex.

Is a quasilinear function always quasiconcave?

A quasilinear function is both quasiconvex and quasiconcave. The graph of a function that is both concave and quasi-convex on the nonnegative real numbers. is a convex set. is strictly quasiconvex.

What is the difference between strictly quasiconcave and strictly quasiconvex?

is strictly quasiconvex. That is, strict quasiconvexity requires that a point directly between two other points must give a lower value of the function than one of the other points does. A quasiconcave function is a function whose negative is quasiconvex, and a strictly quasiconcave function is a function whose negative is strictly quasiconvex.