Cone of positive-semidefinite matrices

[mkoeppe]

We define

• the subspace/submodule of symmetric matrices
• the cone of positive-semidefinite matrices (with ambient = the full matrix space; affine_hull = the above subspace), implementing the ABC for convex sets (ABC for convex sets #31919)
• the cone of positive-definite matrices as its relative_interior.
• additive semigroups/monoids of positive-(semi)definite matrices as intersection of the above with a MatrixSpace

Related:

• sage.tensor.modules: Add SymPowerFreeModule #30095 sage.tensor.modules: Add SymPowerFreeModule
• Category InnerProductSpaces #30218 Category InnerProductSpaces
• ManifoldSubset and ChartFunction objects corresponding to a CVXPY expressions #31981 ManifoldSubset corresponding to a CVXPY Leaf (Variable/Parameter)
• sage.manifolds: Connect to Pymanopt #30495 sage.manifolds: Connect to Pymanopt

Random references:

• Jean-Baptiste Hiriart-Urruty, Jérôme Malick, A Fresh Variational-Analysis Look at the Positive Semidefinite Matrices World, https://hal.archives-ouvertes.fr/hal-00651605/document
• Yuhan Jiang & Bernd Sturmfels, Bad projections of the PSD cone, 2021, https://link.springer.com/article/10.1007/s13348-021-00319-4
• Lek-Heng Lim , Rodolphe Sepulchre, Ke Ye, Geometric Distance Between Positive Definite Matrices of Different Dimensions, https://www.stat.uchicago.edu/~lekheng/work/ellipsoids.pdf
• James Saunderson, Limitations on the expressive power of convex cones without long chains of faces, https://arxiv.org/abs/1902.06401v2

CC: @dimpase @yuan-zhou @orlitzky @kliem

Component: geometry

Author: Matthias Koeppe

Branch/Commit: u/mkoeppe/cone_of_positive_semidefinite_matrices @ 7e16b79

Issue created by migration from https://trac.sagemath.org/ticket/30172

[mkoeppe]

Dependencies: #30095

[mkoeppe]

comment:4

Setting new milestone based on a cursory review of ticket status, priority, and last modification date.

[mkoeppe]

Changed dependencies from #30095 to #31919

[mkoeppe]

Author: Matthias Koeppe

[mkoeppe]

Changed dependencies from #31919 to #31919, #31959

[mkoeppe]

Branch: u/mkoeppe/cone_of_positive_semidefinite_matrices

[sagetrac-git]

Commit: 7e821cf

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

7e821cf

src/sage/geometry/semialgebraic/semidefinite.py: Clarify that all matrices are symmetric

[mkoeppe]

Changed dependencies from #31919, #31959 to #31919, #31959, #31990

[sagetrac-git]

Changed commit from 7e821cf to 59acb68

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

7323b10	ConvexSet_base._test_contains: Only test extension to AA for exact base rings
94e6858	RelativeInterior.ambient, ambient_vector_space, is_universe: New
0c9bc94	ConvexSet_base: Add default implementations of ambient, ambient_dim; add doctests
7d3ae5c	Merge #31959
2b1040c	ConvexSet_base.{an_element,some_elements}, Polyhedron.some_elements, RelativeInterior.some_elements: New
a884517	PolyhedronFace.some_elements: New
a0c6c48	Merge #31990
59acb68	src/sage/geometry/semialgebraic/semidefinite.py: Expand

[mkoeppe]

Changed dependencies from #31919, #31959, #31990 to #31919, #31995, #31959, #31990

[sagetrac-git]

Changed commit from 59acb68 to 47e793d

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

ce91e44	src/sage/geometry/relative_interior.py: Fix doctest output
b19b3cc	Merge #31959
f0e7c58	ambient_vector_space docstring: Fix bad blocks
200d967	ConvexSet_base.ambient doctest: Actually test the method
1c2bef5	Merge #31959
ed898e6	ConvexSet_base.an_element, some_elements: Go through generator method `_some_elements_`, make some_elements return a list
b3ac0f3	ConvexSet_base.an_element, some_elements: Raise NotImplementedError if `_some_elements_` is NotImplemented
7152e1f	ConvexSet_base._test_contains: Test some_elements too
d2dc07e	Merge #31990
47e793d	SemidefiniteMatrices_base.affine_hull: Add doctests

[sagetrac-git]

Changed commit from 47e793d to 5db9a34

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

e270655	Fixing issue when the support is smaller than the dimension.
ad63607	Merge #31995
5db9a34	SemidefiniteMatrices_base: Raise NotImplementedError for non-frield rings; implement _some_elements_

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

44f52b8	sage.geometry.semialgebraic.semidefinite: More examples, code refinements
076d78f	ConvexSet_conic, ConvexSet_closed_conic, ConvexSet_proper_conic: New; move some methods here from ConvexRationalPolyhedralCone, PositiveSemidefiniteMatrices

[sagetrac-git]

Changed commit from 5db9a34 to 076d78f

[sagetrac-git]

Changed commit from 076d78f to 2a17c37

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

2a17c37

Fixup attrcall

[sagetrac-git]

Changed commit from 2a17c37 to 2d890d6

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

991419c	ConvexSet_base._test_convex_set: Do not test _test_as_set_object here
2d890d6	ConvexSet_proper_conic.affine_hull, dim: New

[sagetrac-git]

Changed commit from 2d890d6 to 7e16b79

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

76f8368	sage.geometry.semialgebraic.semidefinite: New
d9be74b	src/sage/geometry/semialgebraic/semidefinite.py: Clarify that all matrices are symmetric
a8f8306	src/sage/geometry/semialgebraic/semidefinite.py: Expand
dd1d87c	SemidefiniteMatrices_base.affine_hull: Add doctests
3d61099	SemidefiniteMatrices_base: Raise NotImplementedError for non-frield rings; implement _some_elements_
9a123f4	sage.geometry.semialgebraic.semidefinite: More examples, code refinements
8dce371	ConvexSet_conic, ConvexSet_closed_conic, ConvexSet_proper_conic: New; move some methods here from ConvexRationalPolyhedralCone, PositiveSemidefiniteMatrices
f7d233a	Fixup attrcall
7e16b79	ConvexSet_proper_conic.affine_hull, dim: New

[mkoeppe]

Changed dependencies from #31919, #31995, #31959, #31990 to none