<p>Hey Evan,</p>
<p>Sorry for the slow response. Things have been busy and I haven't had a chance to look further into this until now.</p>
<p>I wonder, should we even be including an <code>Evaluate()</code> and a <code>Gradient()</code> function in these constraint types? Or instead should we simply be allowing access to the A matrix (however it should be represented) and the b vector, and allow the optimizers to do what they need with those? (This is closer to how the current code works, but the current code has a lot of template specializations that I'd love to do away with, favoring just one non-specialized implementation instead.)</p>
<p>This is simple for the sparse and dense constraints, which can just return A as types <code>arma::sp_mat</code> and <code>arma::mat</code>, respectively, but the linear constraint is a bit harder because we want to represent A as (aa'). We could possibly use the Armadillo type that represents two vectors to be multiplied for this... it'd look something like <code>arma::glue<arma::Col<eT>, arma::Col<eT>, arma::op_multiply></code>. If we return that type for the A matrix and then use that in a larger Armadillo expression, I think we can reasonably expect that the template metaprogramming that evaluates Armadillo expressions will do a decent job.</p>
<p>Is what I'm saying making any sense? Please let me know if not. It has been a while since I have thought about SDPs. My concern though is that ideally we don't want to include to much optimizer-specific logic in the constraints, and it seems like in this case that's what's happening with <code>EvaluateLR()</code> and <code>GradientLR()</code>; if we were to come up with another technique for solving SDPs, like for instance with block diagonal representations instead of low-rank, we would probably need an <code>EvaluateBlockDiagonal()</code> and <code>GradientBlockDiagonal()</code> in each constraint, which I'd like to avoid.</p>
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