Are all quasiconvex functions convex?

Are all quasiconvex functions convex?

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 quasiconcave a concave function?

Similarly, every convex function is quasiconvex. A concave function is quasiconcave. A convex function is quasiconvex. Denote the function by f, and the (convex) set on which it is defined by S.

How do you know if a function is quasiconcave?

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.

Is absolute function convex?

Absolute Value Function is Convex.

Is a linear function quasiconcave?

* A function that is both concave and convex, is linear (well, affine: it could have a constant term). Therefore, we call a function quasilinear if it is both quasiconcave and quasiconvex. Example: any strictly monotone transformation of a linear aTx.

Is a linear function strictly quasiconcave?

In view of Theorem II, a linear function must also be both quasiconcave and quasiconvex, though not strictly so. In the case of concave and convex functions, there is a useful theorem to the effect that the sum of concave (convex) functions is also concave (convex).

Is absolute function concave?

any two points (a, f(a)) and (b, f(b)). For example, the absolute value graph y = |x| has a “pointy smile” ∨ which lies below every secant line crossing the y-axis, and which contains every secant line on one side of the y-axis, so it is concave up.

Is an increasing function quasiconcave?

Any increasing or decreasing function is both quasiconvex and quasiconcave. Quasiconcavity and Quasiconvexity are global properties of a function. Unlike continuity, differentiability, concavity and convexity (of functions), they are not defined at a point.

What is non-convex function?

A non-convex function is wavy – has some ‘valleys’ (local minima) that aren’t as deep as the overall deepest ‘valley’ (global minimum). Optimization algorithms can get stuck in the local minimum, and it can be hard to tell when this happens.

Is every convex function quasiconvex?

Every convex function is quasiconvex but the converse is not true. A function which is both quasiconvex and quasiconcave is called quasimonotone. Let $f:Sightarrow \\mathbb {R}$ and S is a non empty convex set in $\\mathbb {R}^n$.

Can a monotonic transformation of a convex function be concave?

That is, whether or not a function is concave depends on the numbers which the function assigns to its level curves, not just to their shape. The problem with this is that a monotonic transformation of a concave (or convex) function need not be concave (or convex).

What is quasimonotone function?

A function which is both quasiconvex and quasiconcave is called quasimonotone. Let $f:Sightarrow \\mathbb {R}$ and S is a non empty convex set in $\\mathbb {R}^n$.