How does solving dual vs. primal problem make a perf difference in homogenous self-dual formulation?

[Brandon Lee]

Hi, I have a conceptual question. As you indicate many times in the modeling cookbook, I have noticed that Mosek can achieve meaningful speed-ups when solving the dual problem (SOCP in my case).

My understanding was that Mosek considers both the primal and the dual problems, and minimizes for the duality gap. As such, how does choosing to solve the dual vs. the primal problem make a difference in solver speed?

I appreciate your insights. Thank you.

[Erling D. Andersen]

Assume the problem you input to Mosek has the form

min c'x

st Ax = b

    x \in  K

then all other things being equal Mosek prefers that A has few rows and more columns than rows.

In other words, the performance is dependent on what you input as the primal problem in the sense the speed of the underlying linear algebra is dependent on the input problem.

Therefore, Mosek may also dualize the problem if it guesses that the dual form leads to better performance.