(New) Lifted Formulation for Schubert Problems
We used the (reduced) lifted formulation to compute approximations of the 437 3-planes in complex 9-space which nontrivially meet 4 random real 4-planes and 6 random real 6-planes.
Details of the computation of 437 approximate solutions may be found [here].
The lifted formulation uses fewer variables than the primal-dual formulation, 16 rather than 24.
This improved the computation time roughly by a factor of 5.
The system used is square, so certification would be similar to the technique used in the primal-dual formulation.
We used the (reduced) lifted formulation to compute approximations of the 28,490 3-planes in complex 10-space which nontrivially meet 3 random real 5-planes and 12 random real 7-planes.
Details of the computation of 28,490 approximate solutions may be found [here].
The lifted formulation uses fewer variables than the primal-dual formulation, 17 rather than 33.
We did not attempt to solve this problem using the apparently less-efficient primal-dual formulation.
The system is square, so one may certify the approximate solutions using alpha-theory.
We similarly computed approximations of the 128 partial flags of nested 2-planes/4-planes/5-planes in complex 8-space which satisfy various geometric conditions.
Details of the computation of 128 approximate solutions may be found [here].
The lifted formulation again uses fewer variables, 33 rather than 41.
This reduced computation time roughly by 7%.
The system used is square, so certification would be similar to the technique used in the primal-dual formulation.
For comparison, we also include links to the same Schubert problems computed using the primal-dual formulation.
Primal-Dual Formulation for Schubert Problems
We used the primal-dual formulation to compute approximations of the 437 3-planes in complex 9-space which nontrivially meet 4 random real 4-planes and 6 random real 6-planes. Details of the computation and certification of 437 approximate solutions may be found [here].
We similarly computed approximations of the 128 partial flags of nested 2-planes/4-planes/5-planes in complex 8-space which satisfy various geometric conditions. Details of the computation and certification of 128 approximate solutions may be found [here].