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### Deterministic Approach via SOCP

- **Ellipsoid Choice**:  
  Define an ellipsoid \(\mathcal{E}_i\) as:  
  \[
  \mathcal{E}_i = \{\bar{a}_i + P_i u \mid \|u\|_2 \leq 1\} \quad (\bar{a}_i \in \mathbb{R}^n, \, P_i \in \mathbb{R}^{n \times n})
  \]
  - Center: \(\bar{a}_i\)
  - Semi-axes: Determined by singular values/vectors of \(P_i\)

- **Robust Linear Program (LP)**:  
  Minimize \(c^T x\)  
  Subject to \(a_i^T x \leq b_i\) for all \(a_i \in \mathcal{E}_i\), \(i = 1, \ldots, m\)

- **Equivalent Second-Order Cone Program (SOCP)**:  
  Minimize \(c^T x\)  
  Subject to \(\bar{a}_i^T x + \|P_i^T x\|_2 \leq b_i\), \(i = 1, \ldots, m\)

- **Note**:  
  Follows from \(\sup_{\|u\|_2 \leq 1} (\bar{a}_i + P_i u)^T x = \bar{a}_i^T x + \|P_i^T x\|_2\)

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*Source: A. d'Aspremont. Convex Optimization M2. (Slide 58/67)*
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