What is the difference between the First order condition for testing whether a differentiable function is convex and case 2 whether a adifferentiable function is quasiconvex?
Both State the same thing i feel. It is stated in Boyd & Vandenberghe's "Convex Optimization" that if f:Rn→Rf:Rn→R is differentiable, then ff is quasiconvex if and only if
f(y)≤f(x)⟹∇f(x)T(y−x)≤0f(y)≤f(x)⟹∇f(x)T(y−x)≤0, ∀x,y∈domf,∀x,y∈domf, where domfdomf denotes the domain of f.
Which i understand is the same for convex.
Part 2 of my question: How in general other than this do we test if a function is quasiconvex.I read somewhere that the sublevel sets have to be finite or something for the sublevel sets to be convex.Could you give an intuition behind quasiconvex functions?