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]

Description changed:

--- 
+++ 
@@ -1,3 +1,13 @@
-This is an nonpolyhedral convex cone.
+We define
+- the subspace 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 (#31919)
+- the cone of positive-definite matrices as its `relative_interior`.
+
+Related: 
+- #30095 `sage.tensor.modules`: Add `SymPowerFreeModule`
+- #30218 Category `InnerProductSpaces`
+- #31981 `ManifoldSubset` corresponding to a CVXPY `Leaf` (`Variable`/`Parameter`)
+- #30495 `sage.manifolds`: Connect to Pymanopt
 
 
+

[mkoeppe]

Description changed:

--- 
+++ 
@@ -9,5 +9,10 @@
 - #31981 `ManifoldSubset` corresponding to a CVXPY `Leaf` (`Variable`/`Parameter`)
 - #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
 
-

[mkoeppe]

Author: Matthias Koeppe

[mkoeppe]

Description changed:

--- 
+++ 
@@ -10,9 +10,7 @@
 - #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
+- 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
+- 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
 

[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