Generalize ReHLine to solve quadratic problems with linear constraints #6
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ReHLine is designed to solve problems of the kind
$$\min_{\mathbf{\beta} \in \mathbb{R}^d} \frac{1}{2}||\beta||^2 + \sum_{i=1}^n L_i(\mathbf{x}_i^T \mathbf{\beta}) (*)$$ $\mathbf{Aw} + \mathbf{b} \geq \mathbf{0}$ where $L_i$ are PLQ convex loss functions.
s.t.
This PR aims to add a straightforward generalization (for details, see the attachment) to the ReHLine algorithm to solve problems the kind
$$\min_{\mathbf{\beta} \in \mathbb{R}^d} \frac{1}{2}\mathbf{\beta}^T \mathbf{P} \mathbf(\beta) - \mathbf{\mu}^T \mathbf{\beta} + \sum_{i=1}^n L_i(\mathbf{x}_i^T \mathbf{\beta})$$ $\mathbf{Aw} + \mathbf{b} \geq \mathbf{0}$ where $L_i$ are PLQ convex loss functions, and $\mathbf{P}$ is a PSD matrix.
s.t.
It is of course possible to transform the above problem to the original problem (*) using linear transformations. However, there are cases where that might not be desirable. For example, instead of running cholesky decomposition one might have a very fast way to solve$\mathbf{Px} = \mathbf{y}$ (which is intrinsic to how the modified ReHLine algorithm runs), e.g. portfolio optmization with factor modelling.
PR_details.pdf