WIP: use new primal-dual format (linear objectives only)

[chriscoey]

this has probably been the hardest part so far but finally it seems to be working. this PR will make it much easier to hook up to MOI by allowing constraint cones rather than only variable cones, and will reduce the size of the linear systems that need to be solved so will improve performance.

replaces #60
fixes #29
see discussion in #29 for evolving ideas on the new format

• load function, check rank conditions
• initial iterate (using adapted cvxopt technique)
• use cvxopt convergence checks
• update examples
• update tests
• update getter/setter functions
• modify convergence checks using alfonso code ideas
• improve linear system solve
• try modified cvxopt initial iterate technique for HSDE
• construct certificates
• implement hessian-vector products
• pull out the initial iterate finding into separate function
• reduce allocs
• add tests with constraint cones
• debug failure on caprasse example
• debug failure on robinson and rosenbrock examples (primal and dual values very inconsistent)
• cleanup code and add comments

[chriscoey]

working on improving the linear system solve scheme. probably will try the double QR method that CVXOPT uses. in Julia we have the advantage that if A' is sparse, qr(A') will dispatch to SuiteSparse's sparse QR, otherwise it will use LAPACK's dense QR. CVXOPT only uses dense QR.

if I can get the rank of A from qr(A') then that kills another bird with the same stone, namely checking the rank condition for A in the load_data! function. Plus rank(A) currently fails if A is sparse because rank tries to do an SVD and there's no sparse SVD.

[chriscoey]

getting close! should be able to merge in a day or two, then I can work on MOI integration #56 and have access to more tests

[chriscoey]

I have reduced allocations down about as much as possible. there is still a big performance gap between master and this PR. some slowdown is expected because all of the current examples are formulated with variable cones, so the old format was ideal for them. I should be able to bridge that gap more by being more clever about matrix types (eg allowing uniformscaling / identity-like matrices for G, not converting sparse to dense in some places).

[blegat]

Didn't know about this macro. What is the difference between @. a = b and a .= b ?

[chriscoey]

No difference. Better when there's a minus sign though, like on 476, it seems.

[blegat]

Better in which sense ? Less time ? Less allocation ?

[chriscoey]

Allocations

[blegat]

Then it might not allocate the intermediate vector and directly store it in the result, nice !