Add support for DD and SDD matrix and SOS

[tcunis]

1. Add support for DD and SDD matrix cones to sdpsol
2. Add support for DSOS and SDSOS polynomial cones to sossol (and derived sum-of-squares solvers)

Discussed in https://github.com/iFR-ACSO/casos/discussions/72

^{Originally posted by **tcunis** January 16, 2024} The interface for convex sum-of-squares problems (`sossol`) should support convex cones (e.g., Lorentz, SDP) for decision variables which are of degree zero. We want to support the following cones: 1. Convex cones of scalars, vectors, and vectorized matrices in `casos.conic` and `casos.sdpsol`: - `lin` (previously `l`): element-wise inequalities - `lor`, `quad`, or `soc` (previously `q`): Lorentz (quadratic, second-order) cone - `rot` (previously `r`): Rotated Lorentz cone - `psd` (previously `s`): Cone of positive semi-definite matrices - solver-specific cones, e.g., for MOSEK: - `pow` : Power cone - `exp` : Exponential cone 3. Derived cones of vectors and vectorized matrices in `casos.sdpsol`, in addition to those above: - `dd` : Cone of diagonally-dominant matrices (relax to linear inequalities) - `sdd` : Cone of scaled diagonally-dominant matrices (relax to second-order cone constraints) - further derived cones, provided the conic solver supports the base cone; e.g., for MOSEK: - `geo` : Geometric cone (relax to EXP) 4. Polynomial cones in `casos.sossol`: - `lin` (previously `l`): coefficient-wise inequalities (reduces to element-wise inequalities for entries of degree zero) - `sos` (previously `s`): cone of sum-of-squares polynomials (relax to PSD) - `msos` : cone of matrix sum-of-squares polynomials (relax Kronecker product to PSD) - `dsos` : cone of diagonally-dominant sum-of-squares polynomials (relax to DD) - `sdsos` : cone of scaled diagonally-dominant sum-of-squares polynomials (relax to SDD) - any of the cones above, provided the entries are of degree zero and cones are supported by the SDP solver

[renatoloureiro]

To keep track of the developments, the task list with the completed and not completed tasks is presented below

• [x] Diagonally dominant cone (DD)
• [x] Scaled diagonally dominant cone (SDD)
• [x] Diagonally dominant sum-of-squares polynomials (DSOS)
• [x] Scaled diagonally dominant sum-of-squares polynomials (SDSOS)
• [x] Add a demo to compare the solution of the SDPs using PSD or the relaxations, i.e. DD or SDD.
• [x] Add a demo to compare the solution of the sum-of-squares program using SOS, DSOS, and SDSOS.
• [x] Create a test file to check for bugs, i.e., this test file should verify all possible situations, e.g., when all of the cones are used.