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### Optimal and Locally Optimal Points

- **Feasibility**:  
  \( x \) is feasible if \( x \in \text{dom } f_0 \) and it satisfies the constraints.

- **Optimality**:  
  A feasible \( x \) is optimal if \( f_0(x) = p^\star \); \( X_{\text{opt}} \) is the set of optimal points.

- **Local Optimality**:  
  \( x \) is locally optimal if there is an \( R > 0 \) such that \( x \) is optimal for:
  \[
  \begin{align*}
  \text{minimize (over } z) & \quad f_0(z) \\
  \text{subject to} & \quad f_i(z) \leq 0, \quad i = 1, \ldots, m, \\
  & \quad h_i(z) = 0, \quad i = 1, \ldots, p, \\
  & \quad \|z - x\|_2 \leq R
  \end{align*}
  \]

### Examples (with \( n = 1, m = p = 0 \))

- \( f_0(x) = 1/x, \text{ dom } f_0 = \mathbb{R}_{++}: p^\star = 0 \), no optimal point.
- \( f_0(x) = -\log x, \text{ dom } f_0 = \mathbb{R}_{++}: p^\star = -\infty \).
- \( f_0(x) = x \log x, \text{ dom } f_0 = \mathbb{R}_{++}: p^\star = -1/e, x = 1/e \) is optimal.
- \( f_0(x) = x^3 - 3x, p^\star = -\infty \), local optimum at \( x = 1 \).

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*Source: A. d'Aspremont, Convex Optimization M2.*
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