If sparsity is preserved, optimal preconditioners are available, and inequality constraints can be resolved by a multiscale method (or the number of active constraints is not too large), the overall algorithm can be $\mathcal O(n)$ time and space. Distributing across a parallel machine adds a logarithmic term to time. If enough sparsity is available, or if matrix-free methods are used, on the order of one million degrees of freedom can be solved per core. That puts the problem size for today's largest machines at around one trillion degrees of freedom. Several groups have run PDE simulations at this scale.

Note that it is still possible to use Newton-based optimization with large design spaces, you just need to solve iteratively with the Hessian. There are many approaches to doing so efficiently.

If sparsity is preserved, optimal preconditioners are available, and inequality constraints can be resolved by a multiscale method (or the number of active constraints is not too large), the overall algorithm can be $\mathcal O(n)$ time and space. Distributing across a parallel machine adds a logarithmic term to time. If enough sparsity is available, or if matrix-free methods are used, on the order of one million degrees of freedom can be solved per core. That puts the problem size for today's largest machines at around one trillion degrees of freedom. Several groups have run PDE simulations at this scale.

Note that it is still possible to use Newton-based optimization with large design spaces, you just need to solve iteratively with the Hessian. There are many approaches to doing so efficiently.

So it all depends how you define "standard methods". If your definition includes multilevel structure-preserving methods, then extremely large problems are tractable. If your definition is limited to unstructured dense methods, the feasible problem sizes are much smaller because the algorithms are not "scalable" in either time or space.