Time - Complexity Convex Optimization and Eigen Decomposition Say I had the choice of choosing one out of the following two optimization problems which I could use to solve my problem. Which choice is the fastest? How much of a trade-off would it be?  Is the improvement in speed by many factors!?


Minimizing a convex function $L(X)$ in one matrix variable with orthogonality constraints over the matrix-essentially in my case this ends up to solving an eigen-decomposition.
Minimizing the same convex function $L(X)$ with linear constraints in $X$.


I know that 2.) should be faster. But what is the direction of work I need to do- to compare the improvement in speed-especially in terms of using the fastest available eigen solver for 1.)-what would be the corresponding fastest approach to solve 2.)?
Regarding the eigen-decomposition, often it is enough to just compute the first few eigenvalue/eigenvector pairs. In the case, Krylow-subspace methods can be blazingly fast.

For the convex minimization under linear constraints, the (preconditioned) Uzawa iteration is a typical Krylow-subspace method, which is often not bad. There exist probably more advanced algorithms for this problem now, but http://scicomp.stackexchange.com would be a better place to ask about these.  Also Schur-complement methods, or just parameterization of the linear sub-space might work fine too. And in case the linear constraints is actually a bunch of linear inequalities, a state of the art linear/convex programming solver might be the way to go.