Quasiconvex functions

A function \(f\) is quasiconvex iff \(\mathbf{dom} f\) is convex and for any \(x,y \in \mathbf{dom} f\) and \(0\leq \theta \leq 1\)
\[  f(\theta x + (1-\theta) y) \leq \max\{ f(x), f(y) \}\]  A continuous function \(f: \mathbf{R} \rightarrow \mathbf{R}\) is quasiconvex iff at least one of the following conditions holds

• \(f\) is nondecreasing
• \(f\) is nonincreasing
• there is a point \(c\in \mathbf{dom} f\) such that for \(t \leq c\), \(f\) is nonincreasing, and for \(t \ge c\), \(f\) is nondecreasing.

First order conditions

A continuous differentiable function \(f: \mathbf{R}^n \rightarrow \mathbf{R}\) is quasiconvex iff \(\mathbf{dom} f\) is convex and for all \(x,y\in \mathbf{dom}f\) 

\[ f(y) \leq f(x) \Rightarrow \nabla f(x)^T (y-x) \leq 0 \]

Operations that preserve quasiconvexity

• nonnegative weighted maximum
  The function \[ f(x) = \sup_{y\in C} (w_y g_y(x) ) \]  where \(w_y \ge 0\) and \(g_y(x)\) is a parameterized family of quasiconvex functions.
• composition
• if \(g\) is quasiconvex and \(h\) is nondecreasing then, \(f = h \circ g\) is quasiconvex
• composition of a quasiconvex function with an affine or linear-fractional transformation yields a quasiconvex function.
• minimization
  if \(f(x,y)\) is quasiconvex jointly in \(x\) and \(y\) and \(C\) is a convex set, then the function
  \[ g(x) = \inf_{y\in C} f(x,y)\] is quasiconvex.