update termination message for `NearOptimal` status

[lkapelevich]

optimal solution found; terminating
can be confusing

[odow]

I came here to say this:

julia> using JuMP, Hypatia

julia> begin
           model = Model(Hypatia.Optimizer)
           @variable(model, 0 <= x <= 1)
           @variable(model, 0 <= y <= 1)
           @objective(model, Min, -x)
           @constraint(model, x^2 <= y)
           @constraint(model, -x^2 + 1 >= y)
           @constraint(model, x^2 + (y - 0.5)^2 <= 1.0)
           optimize!(model)
       end

 iter        p_obj        d_obj |  abs_gap    x_feas    z_feas |      tau       kap        mu | dir_res     prox  step     alpha
    0  -1.6019e-01  -5.9943e+00 | 1.00e+01  4.57e-01  4.20e-01 | 1.00e+00  1.00e+00  1.00e+00 |
    1  -3.8427e-01  -1.3948e+00 | 2.04e+00  7.46e-02  6.86e-02 | 1.22e+00  1.29e-01  2.00e-01 | 8.9e-16  8.2e-01  co-a  8.00e-01
    2  -7.1577e-01  -9.2310e-01 | 3.47e-01  1.62e-02  1.49e-02 | 1.13e+00  4.00e-02  3.57e-02 | 1.1e-16  8.5e-01  co-a  8.00e-01
    3  -7.0013e-01  -7.2075e-01 | 3.86e-02  1.52e-03  1.40e-03 | 1.20e+00  2.56e-03  3.79e-03 | 5.8e-15  9.2e-01  co-a  9.00e-01
    4  -7.0676e-01  -7.1014e-01 | 5.04e-03  2.57e-04  2.36e-04 | 1.07e+00  4.87e-04  5.05e-04 | 1.6e-15  6.4e-01  co-a  8.50e-01
    5  -7.0700e-01  -7.0751e-01 | 7.60e-04  3.86e-05  3.55e-05 | 1.06e+00  6.67e-05  7.55e-05 | 2.8e-14  5.4e-01  co-a  8.50e-01
    6  -7.0709e-01  -7.0717e-01 | 1.09e-04  6.04e-06  5.55e-06 | 1.02e+00  1.03e-05  1.09e-05 | 8.3e-14  3.3e-01  co-a  8.50e-01
    7  -7.0711e-01  -7.0711e-01 | 1.07e-05  6.18e-07  5.68e-07 | 9.98e-01  1.03e-06  1.06e-06 | 3.6e-13  3.0e-01  co-a  9.00e-01
    8  -7.0711e-01  -7.0711e-01 | 5.21e-07  3.18e-08  2.92e-08 | 9.71e-01  4.89e-08  5.17e-08 | 2.0e-12  8.8e-01  co-a  9.50e-01
    9  -7.0711e-01  -7.0711e-01 | 9.85e-08  6.70e-09  6.16e-09 | 9.20e-01  1.00e-08  9.79e-09 | 3.9e-11  3.9e-01  co-a  8.00e-01
   10  -7.0711e-01  -7.0711e-01 | 8.77e-08  6.06e-09  5.57e-09 | 9.16e-01  9.63e-09  8.77e-09 | 6.1e-11  2.6e-02  co-a  1.00e-01
   11  -7.0711e-01  -7.0711e-01 | 8.34e-08  5.76e-09  5.29e-09 | 9.16e-01  9.10e-09  8.34e-09 | 3.9e-12  1.0e-03  co-a  5.00e-02
   12  -7.0711e-01  -7.0711e-01 | 8.25e-08  5.70e-09  5.24e-09 | 9.16e-01  9.01e-09  8.25e-09 | 6.4e-13  8.8e-06  co-a  1.00e-02
   13  -7.0711e-01  -7.0711e-01 | 8.17e-08  5.64e-09  5.19e-09 | 9.16e-01  8.92e-09  8.17e-09 | 6.5e-15  7.9e-08  co-a  1.00e-02
   14  -7.0711e-01  -7.0711e-01 | 8.13e-08  5.61e-09  5.16e-09 | 9.16e-01  8.88e-09  8.13e-09 | 1.1e-16  5.8e-09  co-a  5.00e-03
   15  -7.0711e-01  -7.0711e-01 | 8.12e-08  5.61e-09  5.16e-09 | 9.16e-01  8.87e-09  8.12e-09 | 1.1e-16  5.4e-09  co-a  1.00e-03
   16  -7.0711e-01  -7.0711e-01 | 8.11e-08  5.60e-09  5.15e-09 | 9.16e-01  8.86e-09  8.11e-09 | 5.6e-17  8.4e-09  co-a  1.00e-03
trying combined without adjustment
trying centering with adjustment
   17  -7.0711e-01  -7.0711e-01 | 8.11e-08  5.60e-09  5.15e-09 | 9.16e-01  8.86e-09  8.11e-09 | 1.8e-16  8.6e-09  ce-a  1.00e+00
trying combined without adjustment
trying centering with adjustment
   18  -7.0711e-01  -7.0711e-01 | 8.11e-08  5.60e-09  5.15e-09 | 9.16e-01  8.86e-09  8.11e-09 | 5.6e-17  7.8e-09  ce-a  1.00e+00
trying combined without adjustment
trying centering with adjustment
   19  -7.0711e-01  -7.0711e-01 | 8.11e-08  5.60e-09  5.15e-09 | 9.16e-01  8.86e-09  8.11e-09 | 5.6e-17  7.8e-09  ce-a  1.00e+00
slow progress in consecutive iterations; terminating
optimal solution found; terminating

status is NearOptimal after 19 iterations and 0.013 seconds

I get:

julia> solution_summary(model)
* Solver : Hypatia

* Status
  Result count       : 1
  Termination status : ALMOST_OPTIMAL
  Message from the solver:
  "NearOptimal"

* Candidate solution (result #1)
  Primal status      : NEARLY_FEASIBLE_POINT
  Dual status        : NEARLY_FEASIBLE_POINT
  Objective value    : -7.07107e-01
  Dual objective value : -7.07107e-01

* Work counters
  Solve time (sec)   : 1.35140e-02
  Barrier iterations : 19

but the solution is strictly primal feasible:

julia> primal_feasibility_report(model)
Dict{Any, Float64}()

Seems like the correct solution to me:

julia> value(x), value(y)
(0.7071067705995698, 0.4999999983377382)