Minimizing a linear function on a strictly convex set.

Question

All the theorems that I know considering the uniqueness of a solution to a minimization/maximization problem requires the strict convexity/strict concavity of the objective function.

But consider the following problem: $$\min_{\substack{x \in \mathbb{R}^n_+ \\ \text{s.t. } f(\mathbf{x})\geq y}}\mathbf{w} \cdot \mathbf{x}$$

Where $\mathbf{w} \in \mathbb{R}^n_{++}$ and $y \in \mathbb{R}_+$ and $f$ is strictly increasing and strictly quasi-concave. Then for a given $y$, the constraint set should be strictly convex so intuitively, this problem has to have a unique solution (it has a solution). Can anybody help me prove that.

Or better, is there a general theorem which tells that whenever you minimize or maximize a linear function on a strictly convex set, the solution has to be unique?

Answer 1

You could prove it along this lines:

Suppose, $x \ne y$ are both feasible and $w \cdot x = w \cdot y$. Then, since the feasible set is strictly convex, $x/2 + y/2$ lies in the interior of the feasible set. Set $z = x/2 + y/2 - \varepsilon \, w$ for small $\varepsilon > 0$. This shows $z$ is feasible and $w \cdot z < w \cdot x$.

Hence, the minimizer is unique.