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@jemoka / Jemoka Knowledge Base / wiki/concepts/worst_case_robust_least_squares.md
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--- title: "worst-case robust least-squares" type: concept source: https://www.jemoka.com/posts/kbhworst_case_robust_least_squares/ confidence: high status: active --- Consider some kind of ellipsoid where your data is constrained: \begin{align} \mathcal{A} = \qty {\bar{A} + u_1 A_1 + \dots + u_{p} A_{p} \mid \norm{u}_{2} \leq 1} \end{align} You can form the “worst-case robust least squares”: \begin{align} \min \text{sup}_{A \in \mathcal{A}} \norm{A x - b}_{2}^{2} = \min \text{sup}_{\norm{u}_{2} \leq } \norm{P\qty(x) u + q\qty(x)}_{2}^{2} \end{align} This is usually a minimax problem, but taking the dual of the inner maximize thing turns out has zero duality gap.