Semidefinite Programs (Experimental)

A semidefinite cone constraint requires a symmetric matrix to be positive semidefinite. In ConicIP, matrices are stored in a vectorized form using vecm and mat.

Vectorization Convention

The cone specification ("S", k) describes a semidefinite cone where k = n(n+1)/2 is the dimension of the vectorized representation of an n × n symmetric matrix.

• vecm(Z) vectorizes a symmetric matrix, scaling off-diagonal entries by √2 so that dot(vecm(X), vecm(Y)) == tr(X*Y).
• mat(x) reconstructs the symmetric matrix from its vectorized form.

Example: Projection onto the PSD Cone

Project the diagonal matrix diag(1, 1, 1, -1, -1, -1) onto the cone of positive semidefinite matrices. The expected result clips the negative eigenvalues to zero: diag(1, 1, 1, 0, 0, 0).

using ConicIP, SparseArrays, LinearAlgebra

# 6×6 matrix → vectorized dimension k = 6*7/2 = 21
k = 21
Q = sparse(1.0I, k, k)
target = diagm(0 => [1.0, 1, 1, -1, -1, -1])
c = ConicIP.vecm(target)

A = sparse(1.0I, k, k)
b = zeros(k)
cone_dims = [("S", k)]

sol = conicIP(Q, c, A, b, cone_dims; verbose=false, optTol=1e-7)
sol.status

:Optimal

Reconstruct the matrix from the solution and check its eigenvalues:

result = ConicIP.mat(sol.y)
round.(eigvals(Symmetric(result)), digits=4)

6-element Vector{Float64}:
 0.0
 0.0
 0.0
 1.0
 1.0
 1.0

The negative eigenvalues have been projected to (approximately) zero.

Understanding vecm and mat

Let's see how the vectorization works on a small example:

X = [1.0 2.0 3.0;
     2.0 5.0 6.0;
     3.0 6.0 9.0]

v = ConicIP.vecm(X)

6-element Vector{Float64}:
 1.0
 2.8284271247461903
 4.242640687119286
 5.0
 8.485281374238571
 9.0

The vector v has length n(n+1)/2 = 6. Off-diagonal entries are scaled by √2:

round.(v, digits=4)

6-element Vector{Float64}:
 1.0
 2.8284
 4.2426
 5.0
 8.4853
 9.0

Reconstruct the original matrix:

X_recovered = ConicIP.mat(v)
round.(X_recovered, digits=4)

3×3 Matrix{Float64}:
 1.0  2.0  3.0
 2.0  5.0  6.0
 3.0  6.0  9.0