Fail with colinearity between objective and constraint

[odow]

This should be a trivial problem to solve, but it fails line search.

See #57

julia> using JuMP, AmplNLWriter, Uno_jll

julia> model = Model(
           () -> AmplNLWriter.Optimizer(Uno_jll.amplexe; directory = "/tmp")
       )
A JuMP Model
├ solver: AmplNLWriter
├ objective_sense: FEASIBILITY_SENSE
├ num_variables: 0
├ num_constraints: 0
└ Names registered in the model: none

julia> @variable(model, x[1:2])
2-element Vector{VariableRef}:
 x[1]
 x[2]

julia> @constraint(model, sum(x) >= 1)
x[1] + x[2] ≥ 1

julia> @objective(model, Min, sum(x))
x[1] + x[2]

julia> optimize!(model)
Original model /tmp/model.nl
2 variables, 1 constraints
Reformulated model /tmp/model.nl_scaled_equalityconstrained_boundrelaxed
3 variables, 1 constraints

Used overwritten options:
- LS_backtracking_ratio = 0.5
- LS_min_step_length = 5e-7
- LS_scale_duals_with_step_length = yes
- armijo_decrease_fraction = 1e-8
- barrier_damping_factor = 1e-5
- barrier_tau_min = 0.99
- constraint_relaxation_strategy = feasibility_restoration
- filter_beta = 0.99999
- filter_fact = 1e4
- filter_gamma = 1e-8
- filter_type = standard
- filter_ubd = 1e4
- globalization_mechanism = LS
- globalization_strategy = waechter_filter_method
- l1_constraint_violation_coefficient = 1000.
- linear_solver = MUMPS
- loose_tolerance = 1e-6
- loose_tolerance_consecutive_iteration_threshold = 15
- progress_norm = L1
- protect_actual_reduction_against_roundoff = yes
- residual_norm = INF
- scale_functions = yes
- sparse_format = COO
- subproblem = primal_dual_interior_point
- switch_to_optimality_requires_linearized_feasibility = no
- switching_delta = 1
- tolerance = 1e-8

┌───────┬─────────┬────────────────┬─────────────┬───────┬────────────────┬──────────────┬─────────────┬───────────────┬──────────────┬─────────────────┬────────────────────────┐
│ iter  │ LS iter │ barrier param. │ step length │ phase │ regularization │ step norm    │ objective   │ primal feas.  │ stationarity │ complementarity │ status                 │
├───────┼─────────┼────────────────┼─────────────┼───────┼────────────────┼──────────────┼─────────────┼───────────────┼──────────────┼─────────────────┼────────────────────────┤
│ 0     │ -       │ -              │ -           │ OPT   │ -              │ -            │ 0           │ 1.01          │ 0            │ 0.01            │ initial point          │
├───────┼─────────┼────────────────┼─────────────┼───────┼────────────────┼──────────────┼─────────────┼───────────────┼──────────────┼─────────────────┼────────────────────────┤
│ 1     │ 1       │ 0.1            │ 1           │ OPT   │ 0              │ 9            │ 0           │ 10.01         │ 1.01         │ 0.0901          │ accepted (h-type)      │

│ -     │ 21      │ -              │ 9.53674e-07 │ -     │ -              │ 3.57589e-07  │ 2.7705      │ 2.02505e-13   │ -            │ -               │ rejected (restoration) │
├───────┼─────────┼────────────────┼─────────────┼───────┼────────────────┼──────────────┼─────────────┼───────────────┼──────────────┼─────────────────┼────────────────────────┤
│ iter  │ LS iter │ barrier param. │ step length │ phase │ regularization │ step norm    │ objective   │ primal feas.  │ stationarity │ complementarity │ status                 │
├───────┼─────────┼────────────────┼─────────────┼───────┼────────────────┼──────────────┼─────────────┼───────────────┼──────────────┼─────────────────┼────────────────────────┤
│ -     │ -       │ -              │ -           │ -     │ -              │ -            │ -           │ -             │ -            │ -               │ LS failed              │
└───────┴─────────┴────────────────┴─────────────┴───────┴────────────────┴──────────────┴─────────────┴───────────────┴──────────────┴─────────────────┴────────────────────────┘

Uno 1.1.0 (LS feasibility_restoration waechter_filter_method primal_dual_interior_point)
Fri Nov  1 13:26:48 2024
────────────────────────────────────────
Status:					Failed with suboptimal point
Objective value:			2.770503
Primal feasibility:			1.412204e-13
┌ Stationarity residual:		1.624039
└ Complementarity residual:		5.085093
┌ Feasibility stationarity residual:	1.624039
└ Feasibility complementarity residual:	5.085093
┌ Infeasibility measure:		1.412204e-13
│ Objective measure:			2.770503
└ Auxiliary measure:			-8.592922e-05
Primal solution:			-2506.469 2509.24 2.770503 
┌ Constraint multipliers:		1.624039 
│ Lower bound multipliers:		0 0 2.872117 
└ Upper bound multipliers:		0 0 0 
┌ Constraint feasibility multipliers:	5.393307e-09 
│ Lower bound feasibility multipliers:	0 0 0.01292776 
└ Upper bound feasibility multipliers:	0 0 0 
Objective multiplier:			0
CPU time:				0.992318s
Iterations:				1362
Objective evaluations:			13369
Constraints evaluations:		13442
Objective gradient evaluations:		1671
Jacobian evaluations:			1671
Hessian evaluations:			1669
Number of subproblems solved:		1669

shell> cat /tmp/model.nl
g3 1 1 0
 2 1 1 0 0 0
 0 1
 0 0
 0 0 0
 0 0 0 1
 0 0 0 0 0
 2 2
 0 0
 0 0 0 0 0
C0
n0
O0 0
n0
x2
0 0
1 0
r
2 1
b
3
3
k1
1
J0 2
0 1
1 1
G0 2
0 1
1 1

[cvanaret]

This is due to the same problem as #58. The inertia correction with MUMPS_jll.jl fails.
My local uno_ampl with MUMPS 5.7.2 reports at the first iteration:

Original matrix
Dimension: 4, number of nonzeros: 10
m(0, 0) = 0
m(1, 1) = 0
m(2, 2) = 0
m(3, 3) = 0
m(0, 0) = 0
m(1, 1) = 0
m(2, 2) = 100
m(0, 3) = 1
m(1, 3) = 1
m(2, 3) = -1

Testing factorization with regularization factors (0, 0)
Expected inertia (3, 1, 0), got (2, 1, 1)
Number of attempts: 1

Matrix is singular
Testing factorization with regularization factors (0.0001, 5.62341e-09)
Dimension: 4, number of nonzeros: 10
m(0, 0) = 0.0001
m(1, 1) = 0.0001
m(2, 2) = 0.0001
m(3, 3) = -5.62341e-09
m(0, 0) = 0
m(1, 1) = 0
m(2, 2) = 100
m(0, 3) = 1
m(1, 3) = 1
m(2, 3) = -1

Factorization was a success

while Uno_jll.jl and MUMPS_jll.jl produce:

Original matrix
Dimension: 4, number of nonzeros: 10
m(0, 0) = 0
m(1, 1) = 0
m(2, 2) = 0
m(3, 3) = 0
m(0, 0) = 0
m(1, 1) = 0
m(2, 2) = 100
m(0, 3) = 1
m(1, 3) = 1
m(2, 3) = -1

Testing factorization with regularization factors (0, 0)
Inertia is good

which is wrong because the matrix is singular.

Again, I have no idea why that happens.
#61 should fix it.