I have derived an optimization objective of the form $$ f(x) = \sum_{i,j} C_{ij}\min(x_i, x_j), s.t. g(x) \geq 0 $$ where $C \in \mathcal{R}^{N \times N}$ is a positive definite matrix, and $x \in \mathcal{R}^{N}$ is a vector where each element $ x_i \geq 1 $. Additionally, $ g(x) \geq 0 $ constrains the solution $ x $ to be inside a convex feasible set.

Is there a way to prove (or disprove) that $ f $ is convex? A simple gradient descent on a test example appear to always converge to the same optimality.

I was able to prove this objective is bounded with $ f(x) \geq \sum_{i,j} C_{ij} $, and the unconstrained case gives $ \forall I: x_i = 1 $ as the optimal solution.

Update: it should be convex but not strongly convex.

I have derived an optimization objective of the form $$ f(x) = \sum_{i,j} C_{ij}\min(x_i, x_j), s.t. g(x) \geq 0 $$ where $C \in \mathcal{R}^{N \times N}$ is a positive definite matrix, and $x \in \mathcal{R}^{N}$ is a vector where each element $ x_i \geq 1 $. Additionally, $ g(x) \geq 0 $ constrains the solution $ x $ to be inside a convex feasible set.

Is there a way to prove (or disprove) that $ f $ is convex? A simple gradient descent on a test example appear to always converge to the same optimality.

I was able to prove this objective is bounded with $ f(x) \geq \sum_{i,j} C_{ij} $, and the unconstrained case gives $ \forall i: x_i = 1 $ as the optimal solution.

I have derived an optimization objective of the form $$ f(x) = \sum_{i,j} C_{ij}\min(x_i, x_j), s.t. g(x) \geq 0 $$ where $C \in \mathcal{R}^{N \times N}$ is a positive definite matrix, and $x \in \mathcal{R}^{N}$ is a vector where each element $ x_i \geq 1 $. Additionally, $ g(x) \geq 0 $ constrains the solution $ x $ to be inside a convex feasible set.

Is there a way to prove (or disprove) that $ f $ is (strongly) convex? A simple gradient descent on a test example appear to always converge to the same optimality.

I was able to prove this objective is bounded with $ f(x) \geq \sum_{i,j} C_{ij} $, and the unconstrained case gives $ \forall I: x_i = 1 $ as the optimal solution.

I have derived an optimization objective of the form $$ f(x) = \sum_{i,j} C_{ij}\min(x_i, x_j), s.t. g(x) \geq 0 $$ where $C \in \mathcal{R}^{N \times N}$ is a positive definite matrix, and $x \in \mathcal{R}^{N}$ is a vector where each element $ x_i \geq 1 $. Additionally, $ g(x) \geq 0 $ constrains the solution $ x $ to be inside a convex feasible set.

Is there a way to prove (or disprove) that $ f $ is convex? A simple gradient descent on a test example appear to always converge to the same optimality.

I was able to prove this objective is bounded with $ f(x) \geq \sum_{i,j} C_{ij} $, and the unconstrained case gives $ \forall I: x_i = 1 $ as the optimal solution.

Update: it should be convex but not strongly convex.