Consider the optimization problem $max_{x\in X} f(x,k)$, where
- $X\subseteq\mathbb{R}^n$ is compact and convex
- $k$ may be vector valued,
- $f(x,k)$ is differentiable in both parameters and concave.
- (In my specific problem, $f(x,k)=\sum_{i=1}^n k_i \ln x_i$.)
Because the problem is convex, we know that there is always a unique optimum $x^*(k)$, and by Berge's Theorem of the maximum it varies continuously with $k$. My question is whether we know anything about the direction in which the optimum moves?
The Lagrangian approach is useful as long as $X$ can be described by finitely many inequalities (via the Jacobian $\partial x^*/\partial k$), but I get stuck for more complicated sets (e.g. the convex hull of a countable collection of points). Intuitively, it seems that when we "tilt" the level curves of $f$ at $x^*(k)$ along a direction that is contained in all supporting hyperplanes at $x^*(k)$, then we should ``slide along the surface'' in the direction of the tilt... but I'm having trouble formalizing the idea, and therefore evaluating its veracity.