Scaled diagonally dominant programming

Specific Domains Optimization (Mathematical)
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2

If I understand correctly, you’d like to use the scaled diagonally dominant cone without Sum-of-Squares is that correct ?
If that’s the case you can do it as follows:

using JuMP, LinearAlgebra, SumOfSquares
model = Model()
# This reformulates the scaled diagonally dominant cone into many 2 x 2 PSD cones
JuMP.add_bridge(model, SumOfSquares.Bridges.Variable.ScaledDiagonallyDominantBridge)
# These reformulate 2 x 2 PSD cones into rotated second order cones
JuMP.add_bridge(model, SumOfSquares.Bridges.Variable.PositiveSemidefinite2x2Bridge)
JuMP.add_bridge(model, SumOfSquares.Bridges.Constraint.PositiveSemidefinite2x2Bridge)
Q = Symmetric(rand(5, 5))
q = JuMP.vectorize(Q, JuMP.SymmetricMatrixShape(5))
@constraint(model, q in SumOfSquares.ScaledDiagonallyDominantConeTriangle(5))

You can inspect and verify that it was transformed to rotated SOC with

julia> using SCS

julia> set_optimizer(model, SCS.Optimizer)

julia> optimize!(model)
------------------------------------------------------------------
	       SCS v3.2.4 - Splitting Conic Solver
	(c) Brendan O'Donoghue, Stanford University, 2012
------------------------------------------------------------------
problem:  variables n: 30, constraints m: 45
cones: 	  z: primal zero / dual free vars: 15
	  q: soc vars: 30, qsize: 10
settings: eps_abs: 1.0e-04, eps_rel: 1.0e-04, eps_infeas: 1.0e-07
	  alpha: 1.50, scale: 1.00e-01, adaptive_scale: 1
	  max_iters: 100000, normalize: 1, rho_x: 1.00e-06
	  acceleration_lookback: 10, acceleration_interval: 10
	  compiled with openmp parallelization enabled
lin-sys:  sparse-direct-amd-qdldl
	  nnz(A): 80, nnz(P): 0
------------------------------------------------------------------
 iter | pri res | dua res |   gap   |   obj   |  scale  | time (s)
------------------------------------------------------------------
     0| 1.18e+00  1.48e-01  1.74e+00  8.68e-01  1.00e-01  1.27e-02 
    25| 1.29e+14  2.86e+11  1.26e+19  6.32e+18  1.00e-01  2.29e-02 
------------------------------------------------------------------
status:  infeasible
timings: total: 2.29e-02s = setup: 1.93e-03s + solve: 2.09e-02s
	 lin-sys: 2.73e-05s, cones: 1.35e-05s, accel: 4.03e-06s
------------------------------------------------------------------
objective = inf
------------------------------------------------------------------

julia> backend(model)
MOIU.CachingOptimizer
├ state: ATTACHED_OPTIMIZER
├ mode: AUTOMATIC
├ model_cache: MOIU.UniversalFallback{MOIU.Model{Float64}}
│ ├ ObjectiveSense: FEASIBILITY_SENSE
│ ├ ObjectiveFunctionType: MOI.ScalarAffineFunction{Float64}
│ ├ NumberOfVariables: 0
│ └ NumberOfConstraints: 1
│   └ MOI.VectorAffineFunction{Float64} in SumOfSquares.ScaledDiagonallyDominantConeTriangle: 1
└ optimizer: MOIB.LazyBridgeOptimizer{MOIU.CachingOptimizer{SCS.Optimizer, MOIU.UniversalFallback{MOIU.Model{Float64}}}}
  ├ Variable bridges:
  │ ├ SumOfSquares.Bridges.Variable.PositiveSemidefinite2x2Bridge{Float64}
  │ └ SumOfSquares.Bridges.Variable.ScaledDiagonallyDominantBridge{Float64}
  ├ Constraint bridges:
  │ ├ MOIB.Constraint.RSOCtoSOCBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.VectorOfVariables}
  │ └ MOIB.Constraint.VectorSlackBridge{Float64, MOI.VectorAffineFunction{Float64}, SumOfSquares.ScaledDiagonallyDominantConeTriangle}
  ├ Objective bridges: none
  └ model: MOIU.CachingOptimizer
    ├ state: EMPTY_OPTIMIZER
    ├ mode: AUTOMATIC
    ├ model_cache: MOIU.UniversalFallback{MOIU.Model{Float64}}
    │ ├ ObjectiveSense: FEASIBILITY_SENSE
    │ ├ ObjectiveFunctionType: MOI.ScalarAffineFunction{Float64}
    │ ├ NumberOfVariables: 30
    │ └ NumberOfConstraints: 11
    │   ├ MOI.VectorAffineFunction{Float64} in MOI.Zeros: 1
    │   └ MOI.VectorAffineFunction{Float64} in MOI.SecondOrderCone: 10
    └ optimizer: SCS.Optimizer
      ├ ObjectiveSense: unknown
      ├ ObjectiveFunctionType: unknown
      ├ NumberOfVariables: unknown
      └ NumberOfConstraints: unknown

Note that it’s further transformed to second-order-cone by Constraint.RSOCtoSOCBridge because SCS does not support the rotated version.

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