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Optimization with convex constraints and convex objectives; notions related to convex optimization such as sub-gradients, normal cones, separating hyperplanes

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How to find $y_u?$

Thanks to this answer. As $L$ is symmetric, \begin{align} \frac{d (E(S, \textbf{y}))}{d y_u} = 0 + y_l^TL^{l,u} + (L^{u,l}y_l)^T + y_u^TL^{u, u} + (L^{u, u}y_u)^T = 0 \\ 2 y_l^T L^{L, u} + 2y_u^T L^{ …
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How to find $y_u?$

In the paper Semi-supervised learning by mixed label propagation Wei Tong and Rong Jin define $S$ as the similarity(adjacency) matrix $D = \operatorname{diag}(D_1, D_2, \ldots, D_n)$ where $D_i = \sum …
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