# SDPSymmetryReduction

Numerically reduces semidefinite programming problems by exploiting their symmetry. Input is in vectorized standard form

```
sup/inf dot(C,x)
subject to Ax = b,
Mat(x) is positive semidefinite/doubly nonnegative,
```

where `C`

and `b`

are vectors and `A`

is a matrix.

## Installation

Simply run

`pkg> add SDPSymmetryReduction # Press ']' to enter the Pkg REPL mode.`

## Determining an admissible subspace

The function `admPartSubspace`

determines an optimal admissible partition subspace for the problem. This is done using a randomized Jordan-reduction algorithm, and it returns a Jordan algebra (closed under linear combinations and squaring). SDPs can be restricted to such a subspace without changing their optimal value.

Given `C,A`

and `b`

, `admPartSubspace(C,a,b)`

returns a `Partition P`

with `P.n`

giving the number of parts of the partition, and `P.P`

returning an integer valued matrix (same size at `x`

in matrix form) with entries `1,...,n`

defining the partition.

## Block-diagonalizing a Jordan-algebra

The function `blockDiagonalize`

determines a block-diagonalization of a (Jordan)-algebra given by a partition `P`

using a randomized algorithm.

`blockDiagonalize(P)`

returns a real block-diagonalization `blkd`

, if it exists, otherwise `nothing`

.

`blkd.blkSizes`

returns an integer array of the sizes of the blocks.`blkd.blks`

returns an array of length`P.n`

containing arrays of (real) matrices of sizes`blkd.blkSizes`

. I.e.`blkd.blks[i]`

is the image of the basis element`P.P .== i`

.

`blockDiagonalize(P; complex = true)`

returns the same, but with complex valued matrices, and should be used if no real block-diagonalization was found. To use the complex matrices practically, remember that a Hermitian matrix `A`

is positive semidefinite iff `[real(A) -imag(A); imag(A) real(A)]`

is positive semidefinite.

## Example: Theta'-function

Let `Adj`

be an adjacency matrix of an (undirected) graph `G`

. Then the Theta'-function of the graph is given by

```
sup dot(J,X)
subject to dot(Adj,X) = 0,
dot(I,X) = 1,
X is positive semidefinite,
X is entry-wise nonnegative,
```

where `J`

is the all-ones matrix, and `I`

the identity. Then we can exploit the symmetry of the graph and calculate this function by

```
using SDPSymmetryReduction
using LinearAlgebra, SparseArrays
using JuMP, MosekTools
# Theta' SDP
N = size(Adj,1)
C = ones(N^2)
A = hcat(vec(Adj), vec(Matrix(I, N, N)))'
b = [0, 1]
# Find the optimal admissible subspace (= Jordan algebra)
P = admPartSubspace(C, A, b, true)
# Block-diagonalize the algebra
blkD = blockDiagonalize(P, true)
# Calculate the coefficients of the new SDP
PMat = hcat([sparse(vec(P.P .== i)) for i = 1:P.n]...)
newA = A * PMat
newB = b
newC = C' * PMat
# Solve with optimizer of choice
m = Model(Mosek.Optimizer)
# Initialize variables corresponding parts of the partition P
# >= 0 because the original SDP-matrices are entry-wise nonnegative
x = @variable(m, x[1:P.n] >= 0)
@constraint(m, newA * x .== newB)
@objective(m, Max, newC * x)
psdBlocks = sum(blkD.blks[i] .* x[i] for i = 1:P.n)
for blk in psdBlocks
if size(blk, 1) > 1
@constraint(m, blk in PSDCone())
else
@constraint(m, blk .>= 0)
end
end
optimize!(m)
@show termination_status(m)
@show value(newC * x)
```

There are more examples in the folder `examples`

:

`ReduceAndSolveJuMP.jl`

: A more advanced function for fully reducing and solving SDPs with JuMP and Mosek, including support for complex block-diagonalizations.`ErdosRenyiThetaFunction.jl`

: A full example calculating the Theta'-function of Erdos-Renyi graphs.`QuadraticAssignmentProblems.jl`

: Loads a QAP in QAPLib format and then reduces and solves a strong doubly nonnegative relaxation of it.