Integrability of Pseudomonotone Differentiable Maps and the Revealed Preference Problem
JP Crouzeix, Tamás Rapcsák · Journal of convex analysis · 2005
The problem considered is as follows: given C\subset {{\mathbb R}}^n C ⊂ R n and F:C\rightarrow {{\mathbb R}}^n F : C → R n differentiable, find f:C\rightarrow {{\mathbb R}} f : C → R differentiable such that \vert\vert abla f(x)\vert\vert^{-1} abla f(x) = \vert\vert F(x) \vert\vert^{-1}F(x) ∣ ∣ ∇ f ( x ) ∣ ∣ − 1 ∇ f ( x ) = ∣ ∣ F ( x ) ∣ ∣ − 1 F ( x ) for all x\in C x ∈ C . Conditions for f f to be pseudoconvex or convex are given. The results are applied to the differentiable case of the revealed preference problem.