Quasiconvex function

A quasiconvex function is a real-valued function on a convex set whose every sublevel set is convex. If the domain is $D$, this meansis a convex set for every real $\alpha$, including levels for which the set is empty. Equivalently,Boyd and Vandenberghe give these level-set and segment characterizations.^[ref]

To see their equivalence, set $\alpha =\max \{f(x),f(y)\}$. Convexity of that sublevel set puts every convex combination of $x$ and $y$ inside it, proving the inequality. Conversely, if two points lie below the same level, the inequality keeps their connecting segment below that level.

Quasiconvexity is weaker than convexity. An original example is $f(x)=\sqrt{|x|}$ on $\mathbb{R}$. For $\alpha \ge 0$, its sublevel set is $[-{\alpha }^2,{\alpha }^2]$; for $\alpha <0$, it is empty. All are convex, butwhich violates the convex-function inequality. Quasiconvexity limits the height along a segment by the larger endpoint value, rather than by the line joining endpoint values.

A strictly increasing transformation preserves the sublevel ordering and hence quasiconvexity. This can be useful when a complicated objective has simple level sets. Quasiconvex optimization exploits those sets through feasibility tests at trial objective values; it need not treat the objective itself as a convex function.

In one dimension the shape corresponds to the single-minimum convention of unimodality. Flat portions require care: the step function equal to0 for $x\le 0$ and1 for $x>0$ is quasiconvex. Every point strictly to the right of zero is a non-strict local minimizer on a small flat neighborhood, yet its value is not globally minimal. Quasiconvexity alone therefore does not justify replacing every local minimum by a global one.