Any way to achieve the L2 norm of a matrix xx is smaller than 1 in JuMP?

Specific Domains Optimization (Mathematical)
,
2

You’re pretty close, you just needed vec(A):

julia> using JuMP

julia> model = Model()
A JuMP Model
Feasibility problem with:
Variables: 0
Model mode: AUTOMATIC
CachingOptimizer state: NO_OPTIMIZER
Solver name: No optimizer attached.

julia> @variable(model, A[1:3, 1:3])
3×3 Matrix{VariableRef}:
 A[1,1]  A[1,2]  A[1,3]
 A[2,1]  A[2,2]  A[2,3]
 A[3,1]  A[3,2]  A[3,3]

julia> @constraint(model, [1; vec(A)] in SecondOrderCone())
[1, A[1,1], A[2,1], A[3,1], A[1,2], A[2,2], A[3,2], A[1,3], A[2,3], A[3,3]] ∈ MathOptInterface.SecondOrderCone(10)

julia> [1; A]
4×3 Matrix{AffExpr}:
 1       1       1
 A[1,1]  A[1,2]  A[1,3]
 A[2,1]  A[2,2]  A[2,3]
 A[3,1]  A[3,2]  A[3,3]

julia> [1; vec(A)]
10-element Vector{AffExpr}:
 1
 A[1,1]
 A[2,1]
 A[3,1]
 A[1,2]
 A[2,2]
 A[3,2]
 A[1,3]
 A[2,3]
 A[3,3]