Duality gap in Convex.jl

Specific Domains Optimization (Mathematical)
1

Using the SCS solver with Convex.jl, the output of the solver includes in its printed output the duality gap:

------------------------------------------------------------------
	       SCS v3.2.1 - Splitting Conic Solver
	(c) Brendan O'Donoghue, Stanford University, 2012
------------------------------------------------------------------
problem:  variables n: 2701, constraints m: 7851
cones: 	  z: primal zero / dual free vars: 1
	  l: linear vars: 50
	  q: soc vars: 7800, qsize: 2600
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
lin-sys:  sparse-direct-amd-qdldl
	  nnz(A): 15151, nnz(P): 0
------------------------------------------------------------------
 iter | pri res | dua res |   gap   |   obj   |  scale  | time (s)
------------------------------------------------------------------
     0| 4.39e+00  1.00e+00  4.96e+01  2.43e+01  1.00e-01  1.02e-02 
   250| 4.43e-03  7.32e-03  2.04e-01  7.26e+00  1.00e-01  1.34e-01 
   425| 2.05e-04  1.91e-05  1.42e-04  6.96e+00  3.15e-02  2.24e-01 
------------------------------------------------------------------
status:  solved
timings: total: 2.24e-01s = setup: 8.90e-03s + solve: 2.15e-01s
	 lin-sys: 9.77e-02s, cones: 5.55e-02s, accel: 5.25e-03s
------------------------------------------------------------------
objective = 6.961736
------------------------------------------------------------------

Is there a simple way to access the final duality gap programmatically using Convex.jl?

2

You can query the underlying MathOptInterface model. For example, if we copy the tutorial example,

Tutorial code 
# Make the Convex.jl module available
using Convex, SCS

# Generate random problem data
m = 4;  n = 5
A = randn(m, n); b = randn(m, 1)

# Create a (column vector) variable of size n x 1.
x = Variable(n)

# The problem is to minimize ||Ax - b||^2 subject to x >= 0
# This can be done by: minimize(objective, constraints)
problem = minimize(sumsquares(A * x - b), [x >= 0])

# Solve the problem by calling solve!
solve!(problem, SCS.Optimizer; silent_solver = true)

# Check the status of the problem
problem.status # :Optimal, :Infeasible, :Unbounded etc.

# Get the optimum value
problem.optval

We can get the duality gap as:

julia> import MathOptInterface as MOI

julia> MOI.get(problem.model, MOI.ObjectiveValue()) - MOI.get(problem.model, MOI.DualObjectiveValue())
7.153038624529984e-6
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3

Thank you! Of course, this is exactly what I need.

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