[docs] improve the docstring for DUAL_INFEASIBLE

[odow]

x-ref https://discourse.julialang.org/t/approaching-a-linear-program-correctly/127354/15?u=odow

[WalterMadelim]

I think the first statement is aggressive. It asserts more information than what it really proves.
The following example you proposed is a nice one to reflect this point.

import JuMP, MosekTools
model = JuMP.Model(MosekTools.Optimizer) # the primal problem
# `model` admits of a `DUAL_INFEASIBLE` dual problem
# The odds are that Mosek reports `DUAL_INFEASIBLE` before reporting `OPTIMAL`
# In this case, the 1st statement is not accurate, because actually a dual bound exists, see the end line
JuMP.@variable(model, X[1:2, 1:2], PSD)
JuMP.@constraint(model, X[1, 1] == 0)
JuMP.optimize!(model)
JuMP.assert_is_solved_and_feasible(model; allow_local = false)
JuMP.objective_bound(model) # 0.0, indicating that a dual bound (being 0.0) does exist for the problem

[WalterMadelim]

I find something weird, why does Mosek reports slow_progress to this

julia> import JuMP, MosekTools

julia> model = JuMP.Model(MosekTools.Optimizer) # the primal problem
A JuMP Model
├ solver: Mosek
├ objective_sense: FEASIBILITY_SENSE
├ num_variables: 0
├ num_constraints: 0
└ Names registered in the model: none

julia> JuMP.set_silent(model)

julia> JuMP.@variable(model, X[1:2, 1:2], PSD)
2×2 LinearAlgebra.Symmetric{JuMP.VariableRef, Matrix{JuMP.VariableRef}}:
 X[1,1]  X[1,2]
 X[1,2]  X[2,2]

julia> JuMP.@constraint(model, X[1, 1] == 0)
X[1,1] == 0

julia> JuMP.@objective(model, Min, 2 * X[1, 2])
2 X[1,2]

julia> JuMP.optimize!(model)

julia> JuMP.assert_is_solved_and_feasible(model; allow_local = false)
ERROR: The model was not solved correctly. Here is a summary of the solution to help debug why this happened:

* Solver : Mosek

* Status
  Result count       : 1
  Termination status : SLOW_PROGRESS
  Message from the solver:
  "Mosek.MSK_RES_TRM_STALL"

* Candidate solution (result #1)
  Primal status      : UNKNOWN_RESULT_STATUS
  Dual status        : UNKNOWN_RESULT_STATUS
  Objective value    : -1.09732e+01
  Objective bound    : -1.09732e+01
  Relative gap       : 0.00000e+00
  Dual objective value : 0.00000e+00

* Work counters
  Solve time (sec)   : 0.00000e+00
  Simplex iterations : 0
  Barrier iterations : 24
  Node count         : 0

Stacktrace:
 [1] error(s::String)
   @ Base .\error.jl:35
 [2] assert_is_solved_and_feasible(model::JuMP.Model; result::Int64, kwargs::@Kwargs{allow_local::Bool})    
   @ JuMP K:\judepot1114\packages\JuMP\xlp0s\src\optimizer_interface.jl:950
 [3] top-level scope
   @ REPL[15]:1

[WalterMadelim]

Regradless, the theoretical behavior for this SDP problem

@variable(model, X[1:2, 1:2], PSD)
@constraint(model, X[1, 1] == 0)
@objective(model, Min, 2 * X[1, 2])

should be: Its objective_bound = objective_value = 0.
Therefore this SDP essentially has a dual bound (which is not derived from the dual side).
The odds are that a solver reports DUAL_INFEASIBLE to this SDP firstly, which is not incorrect.

[odow]

Your first example did not include the objective.

Here's what I get with the latest JuMP and MosekTools. The important part is objective_bound : NaN. The problem does not have a dual bound.

julia> using JuMP, MosekTools

julia> model = Model(MosekTools.Optimizer)
A JuMP Model
├ solver: Mosek
├ objective_sense: FEASIBILITY_SENSE
├ num_variables: 0
├ num_constraints: 0
└ Names registered in the model: none

julia> set_attribute(model, "fallback", "mosek://solve.mosek.com:30080")

julia> @variable(model, X[1:2, 1:2], PSD)
2×2 LinearAlgebra.Symmetric{VariableRef, Matrix{VariableRef}}:
 X[1,1]  X[1,2]
 X[1,2]  X[2,2]

julia> @constraint(model, X[1, 1] == 0)
X[1,1] = 0

julia> @objective(model, Min, 2 * X[1, 2])
2 X[1,2]

julia> optimize!(model)
Problem
  Name                   :                 
  Objective sense        : minimize        
  Type                   : CONIC (conic optimization problem)
  Constraints            : 1               
  Affine conic cons.     : 0               
  Disjunctive cons.      : 0               
  Cones                  : 0               
  Scalar variables       : 0               
  Matrix variables       : 1 (scalarized: 3)
  Integer variables      : 0               

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries                  : 2                 time                   : 0.00            
Lin. dep.  - tries                  : 1                 time                   : 0.00            
Lin. dep.  - primal attempts        : 1                 successes              : 1               
Lin. dep.  - dual attempts          : 0                 successes              : 0               
Lin. dep.  - primal deps.           : 0                 dual deps.             : 0               
Presolve terminated. Time: 0.00    
Optimizer  - threads                : 1               
Optimizer  - solved problem         : the primal      
Optimizer  - Constraints            : 1               
Optimizer  - Cones                  : 1               
Optimizer  - Scalar variables       : 3                 conic                  : 3               
Optimizer  - Semi-definite variables: 0                 scalarized             : 0               
Factor     - setup time             : 0.00            
Factor     - dense det. time        : 0.00              GP order time          : 0.00            
Factor     - nonzeros before factor : 1                 after factor           : 1               
Factor     - dense dim.             : 0                 flops                  : 7.00e+00        
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  
0   7.1e-01  1.4e+00  1.0e+00  0.00e+00   0.000000000e+00   0.000000000e+00   1.0e+00  0.00  
1   9.7e-02  1.9e-01  1.7e-01  -3.33e-01  -1.238467732e+00  0.000000000e+00   1.4e-01  0.00  
2   2.1e-02  4.1e-02  3.9e-02  -1.05e-01  -1.656080850e+00  0.000000000e+00   2.9e-02  0.00  
3   4.5e-03  8.9e-03  8.7e-03  -4.49e-02  -1.837436144e+00  0.000000000e+00   6.3e-03  0.00  
4   9.6e-04  1.9e-03  1.9e-03  -2.07e-02  -1.923827145e+00  0.000000000e+00   1.4e-03  0.00  
5   2.1e-04  4.1e-04  4.1e-04  -9.61e-03  -1.964762106e+00  0.000000000e+00   2.9e-04  0.00  
6   4.3e-05  8.6e-05  8.6e-05  -4.42e-03  -1.983870012e+00  0.000000000e+00   6.1e-05  0.00  
7   8.9e-06  1.8e-05  1.8e-05  -2.02e-03  -1.992697438e+00  0.000000000e+00   1.3e-05  0.00  
8   1.8e-06  3.6e-06  3.6e-06  -9.14e-04  -1.996740886e+00  0.000000000e+00   2.5e-06  0.00  
9   3.4e-07  6.8e-07  6.8e-07  -4.08e-04  -1.998579228e+00  0.000000000e+00   4.8e-07  0.00  
10  6.0e-08  1.2e-07  1.2e-07  -1.77e-04  -1.999411898e+00  0.000000000e+00   8.5e-08  0.00  
11  1.2e-08  2.4e-08  2.4e-08  -8.73e-05  -1.999827434e+00  0.000000000e+00   1.7e-08  0.00  
12  2.6e-09  5.2e-09  5.2e-09  8.54e-05   -1.999173194e+00  0.000000000e+00   3.7e-09  0.00  
13  5.6e-10  1.1e-09  1.1e-09  -8.99e-04  -2.005449121e+00  0.000000000e+00   8.0e-10  0.00  
14  1.3e-10  2.5e-10  2.5e-10  7.49e-03   -1.954650036e+00  0.000000000e+00   1.8e-10  0.00  
15  3.2e-11  6.4e-11  6.9e-11  -5.51e-02  -2.328648409e+00  0.000000000e+00   4.5e-11  0.00  
16  9.9e-12  2.0e-11  1.5e-11  2.25e-01   -1.198281673e+00  0.000000000e+00   1.4e-11  0.01  
17  2.4e-12  2.3e-11  8.6e-12  -3.02e-01  -6.668106594e+00  0.000000000e+00   3.3e-12  0.01  
18  3.0e-13  7.5e-10  2.3e-13  1.86e-01   -2.845957952e-01  0.000000000e+00   4.3e-13  0.01  
19  2.4e-14  1.3e-09  2.7e-14  3.67e-03   -5.915090097e-01  0.000000000e+00   3.5e-14  0.01  
20  1.6e-15  9.8e-10  2.6e-15  -2.41e-01  -2.572824537e+00  0.000000000e+00   1.7e-15  0.01  
21  3.1e-16  7.9e-09  9.6e-16  -4.30e-01  -1.097317669e+01  0.000000000e+00   2.0e-16  0.01  
22  3.1e-16  7.9e-09  9.6e-16  -4.30e-01  -1.097317669e+01  0.000000000e+00   2.0e-16  0.01  
23  3.1e-16  7.9e-09  9.6e-16  -4.30e-01  -1.097317669e+01  0.000000000e+00   2.0e-16  0.01  
Optimizer terminated. Time: 0.01    


Interior-point solution summary
  Problem status  : UNKNOWN
  Solution status : UNKNOWN
  Primal.  obj: -1.0973176689e+01   nrm: 7e+08    Viol.  con: 7e-08    barvar: 1e-07  
  Dual.    obj: 0.0000000000e+00    nrm: 1e+08    Viol.  con: 0e+00    barvar: 2e+00  

julia> solution_summary(model)
solution_summary(; result = 1, verbose = false)
├ solver_name          : Mosek
├ Termination
│ ├ termination_status : SLOW_PROGRESS
│ ├ result_count       : 1
│ ├ raw_status         : Mosek.MSK_RES_TRM_STALL
│ └ objective_bound    : NaN
├ Solution (result = 1)
│ ├ primal_status        : UNKNOWN_RESULT_STATUS
│ ├ dual_status          : UNKNOWN_RESULT_STATUS
│ ├ objective_value      : -1.09732e+01
│ ├ dual_objective_value : 0.00000e+00
│ └ relative_gap         : NaN
└ Work counters
  ├ solve_time (sec)   : 9.68066e-03
  ├ simplex_iterations : 0
  ├ barrier_iterations : 24
  └ node_count         : 0

[WalterMadelim]

Your first example did not include the objective.

Yes, I was careless. But the meaning is already conveyed.

Additionally, according to my experience, Mosek very frequently give slow_progress.
And SDP is indeed harder to handle. The numerical issues are often severe.
No wonder Gurobi do not support solving SDP problems.

[odow]

I tweaked the wording to be more direct. I don't think we're going to be able to please everyone without a massive amount of detail, but this hopefully suffices for nearly everyone.

[WalterMadelim]

The updated one (c75f852) is good.