\[\min_{x}\; c^{\top}x \quad \text{s.t.}\; Ax = b,\; x \ge 0\]
\[\max_{y}\; b^{\top}y \quad \text{s.t.}\; A^{\top}y \le c\]
\[\mathcal{L}(x,\lambda) = f(x) + \lambda^{\top} g(x)\]
\[\nabla f(x^{*}) + \sum_{i} \lambda_i \nabla g_i(x^{*}) = 0\]
\[\lambda_i\, g_i(x^{*}) = 0,\quad \lambda_i \ge 0\]
\[c^{\top}x \;\ge\; b^{\top}y\]
\[x_{k+1} = x_k - \alpha_k \nabla f(x_k)\]
\[x_{k+1} = x_k - \left[\nabla^{2} f(x_k)\right]^{-1} \nabla f(x_k)\]
\[V(s) = \max_{a}\Big[ r(s,a) + \gamma \sum_{s'} P(s' \mid s,a)\, V(s') \Big]\]
\[x \in \{0,1\}^{n}\]
\[\sum_{j} a_{ij} x_j \le b_i \quad \forall i\]
\[\text{gap} = \frac{z_{\text{UB}} - z_{\text{LB}}}{|z_{\text{UB}}|}\]
Decisions, solved.
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