(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].