Understanding Quasiconvex Functions: Examples and Characteristics

Quasiconvex functions are an interesting subset of functions that offer a broader definition of convexity. Unlike convex functions, which require every level set to be convex, quasiconvex functions only demand that all level sets must be convex sets. This less restrictive condition allows for the presence of certain non-convex behaviors, making them a valuable topic in mathematical optimization and analysis.

Examples of Quasiconvex Functions that are Not Convex

Let's explore a few examples of quasiconvex functions and see why they don't satisfy the stronger condition of convexity.

Piecewise Linear Function

Consider the piecewise linear function defined as:

[ f(x) begin{cases} 1 - x text{if } x leq 1 0 text{if } x 1 end{cases} ]

This function is quasiconvex because all its level sets { x : f(x) leq t } are convex. However, it is not convex since it has a kink at ( x 1 ).

The Function ( f(x) -x )

This linear function is defined as ( f(x) -x ). It is quasiconvex because the level sets { x : f(x) leq t } are intervals, which are convex. However, it is not convex because it lies below the line connecting any two of its points.

The Function ( f(x) x^2 sinleft(frac{1}{x}right) ) for ( x neq 0 ) and ( f(0) 0 )

This function oscillates as ( x ) approaches 0, creating a non-convex shape. The level sets for ( t leq 0 ) are empty, and for ( t 0 ) and ( t 0 ), they are convex. Thus, it is quasiconvex but not convex.

The Function ( f(x) x^4 - 4x^2 )

The function has local minima and maxima, leading to a non-convex shape. For ( x ) in the range ( [-2, 2] ), the function dips below the line connecting points at ( x -2 ) and ( x 2 ), showing it is not convex. However, the level sets are convex.

The Function ( f(x) x^2 - 1 ) for ( x in [-1, 1] )

This function is a downward-opening parabola on the interval ([-1, 1]). Its level sets are intervals, which are convex, but the function itself is not convex over that interval.

Simple and Increasing Functions are Quasiconvex

It is worth noting that any increasing function ( f: mathbb{R} rightarrow mathbb{R} ) is quasiconvex. This is always true when the domain of ( f ) is the real line ( mathbb{R} ) or any subset of it.

Examples:

[ f_1(x) x^3 ] [ f_2(x) min(x, 1) ]

Functions with Higher Dimensional Domains

Quasiconvex functions can also be defined over higher-dimensional domains. Here are a couple of examples:

( f(x, y) sqrt{max(x, y)} )

This function involves the maximum operation over two variables and the square root.

( f(x, y) log(x^2 y^2) )

This function combines the logarithm and power operations over two variables.

These examples illustrate how quasiconvexity allows for certain non-convex behaviors while still ensuring that level sets maintain a convex structure. Understanding these functions can be crucial in various mathematical models and optimization problems, particularly in areas like economics, engineering, and machine learning.