In mathematics, a quasiconvex function is a real-valued function defined on a convex subset of a real vector space, such that for any real number y, the set of points on which the function value is at most y is a convex set. In other words, the inverse image of any set of the form ( − ∞ , y ) {\displaystyle (-\infty ,y)} is a convex set. An equivalent definition is: along any interval in the function domain, the function attains the highest value on one of the endpoints. Quasiconvexity is a more general property than convexity: all convex functions are also quasiconvex, but not all quasiconvex functions are convex. For one-dimensional functions (functions on R), to check graphically whether a function is quasiconvex, move a horizontal line from minus infinity upwards, and verify that, whenever the line intersects the region above the function graph, the intersection is an interval. A quasiconcave function is the negative of a quasiconvex function. In a quasiconcave function, for any real number y, the set of points on which the function value is at least y is convex. Equivalently, along any interval in the function domain, the function attains the lowest value on one of the endpoints. In one dimension, verify that, for any horizontal line that intersects the region below the function graph, the intersection is an interval. Univariate unimodal functions are quasiconvex or quasiconcave, however this is not necessarily the case for functions with multiple arguments. For example, the 2-dimensional Rosenbrock function is unimodal but not quasiconvex and functions with star-convex sublevel sets can be unimodal without being quasiconvex.
Definition and properties A function f : S → R {\displaystyle f:S\to \mathbb {R} } defined on a convex subset S {\displaystyle S} of a real vector space is quasiconvex if for all x , y ∈ S {\displaystyle x,y\in S} and λ ∈ [ 0 , 1 ] {\displaystyle \lambda \in [0,1]} we have
f ( λ x + ( 1 − λ ) y ) ≤ max { f ( x ) , f ( y ) } . {\displaystyle f(\lambda x+(1-\lambda )y)\leq \max {\big \{}f(x),f(y){\big \}}.}
In words, the objective f {\displaystyle f} is quasiconvex if and only if the maximum of f {\displaystyle f} along a straight line between any two end points is never greater than the value at the higher endpoint. Note that the points x {\displaystyle x} and y {\displaystyle y} may be points in n-dimensional space. If the inequality is strict, i.e.
f ( λ x + ( 1 − λ ) y ) < max { f ( x ) , f ( y ) } {\displaystyle f(\lambda x+(1-\lambda )y)<\max {\big \{}f(x),f(y){\big \}}}
for all x ≠ y {\displaystyle x\neq y} and λ ∈ ( 0 , 1 ) {\displaystyle \lambda \in (0,1)} , then f {\displaystyle f} 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.
An alternative way (see introduction) of defining a quasi-convex function f ( x ) {\displaystyle f(x)} is to require that each sublevel set
S α ( f ) = { x ∣ f ( x ) ≤ α } {\displaystyle S_{\alpha }(f)=\{x\mid f(x)\leq \alpha \}}
is a convex set. It follows that for every strictly quasiconvex function, there exist a strictly monotone increasing coordinate transformation m : R → R {\displaystyle m:\mathbb {R} \to \mathbb {R} }
such that m ( f ( x ) ) {\displaystyle m(f(x))} is strictly convex. A quasiconcave function is a function whose negative is quasiconvex, and a strictly quasiconcave function is a function whose negative is strictly quasiconvex. Equivalently a function f {\displaystyle f} is quasiconcave if and only if
f ( λ x + ( 1 − λ ) y ) ≥ min { f ( x ) , f ( y ) } . {\displaystyle f(\lambda x+(1-\lambda )y)\geq \min {\big \{}f(x),f(y){\big \}}.}
A (strictly) quasiconvex function has (strictly) convex lower contour sets, while a (strictly) quasiconcave function has (strictly) convex upper contour sets. Unimodal probability distributions like the Gaussian distribution are common examples of quasi-concave functions that are not concave. A function that is both quasiconvex and quasiconcave is quasilinear, and satisfies
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