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### Local and global optima

- **Statement**: Any locally optimal point of a convex problem is (globally) optimal.

- **Proof Outline**:
  - Assume \( x \) is locally optimal and \( y \) is optimal with \( f_0(y) < f_0(x) \).
  - Local optimality of \( x \) implies the existence of \( R > 0 \) such that for any feasible \( z \) with \( \|z - x\|_2 \leq R \), it follows that \( f_0(z) \geq f_0(x) \).
  - Consider \( z = \theta y + (1 - \theta)x \) where \( \theta = R/(2\|y - x\|_2) \).
    - \(\|y - x\|_2 > R\), hence \( 0 < \theta < 1/2 \).
    - \( z \) is a convex combination of two feasible points, making it feasible.
    - \(\|z - x\|_2 = R/2\).
    - \( f_0(z) \leq \theta f_0(x) + (1 - \theta)f_0(y) < f_0(x) \).
  - This results in a contradiction, proving that \( x \) must be globally optimal.

- **Source**: A. d'Aspremont, Convex Optimization M2, Slide 41/67.
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