Hi Jack,

Sub-optimal termination is printed when the barrier algorithm does not make any (or only very small) progress in the solution value and cannot decrease the dual residual, meaning that although the final convergence criterion is met, the dual infeasibility cannot be reduced. This is often the case for shaky numerics.

You can try to overcome this issue by either reformulating your problem and/or experimenting with the following parameters; this may however result in a performance loss:

$\min_{b_1,b_2} \sum_j (y_j - b_1 - x_j\cdot b_2)^2$

where $$y_j, x_j$$ are constants and $$b_{1},b_2$$ are variables.
You could  formulate it as

$\min_{b_1,b_2} \sum_j (y_j^2 - 2\cdot y_j\cdot b_1 - 2\cdot y_j\cdot x_j \cdot b_2) + \sum_j (b_1 + x_j\cdot b_2)^2$

If none of the above helps, could you then provide an MPS or an LP file of the model, see How do I write an MPS model file from a third-party API and How do I write an "MPS file" for my problem? To make this file available to the community, please have a look at Posting to the Community Forum.

Best regards,
Jaromił

Hi Jaromił,

Thanks for the help. I still get some sub-optimal termination warnings when playing with the various tuning parameters in both the original and re-formulated problem. Below is the MPS output for the following problem (the upper bound on b_2 is found analogously):

\begin{align}
\underset{b_1,b_2}{\min} \ b_2 &\quad s.t. \\
\sum_{j=1}^N (y_j - b_1 -b_2x_j)^2 &\leq \underset{b_1,b_2}{\min} \sum_{j=1}^N (y_j - b_1 -b_2x_j)^2 \\
b_1,b_2 &\in [-5,5]
\end{align}

MPS:

NAME fooOBJSENSE MAXROWS N  OBJ L  qc0     COLUMNS    C0        OBJ       0    C0        qc0       -5.0347716616104652e+00    C1        OBJ       1    C1        qc0       -4.0309243407684274e+00RHS    RHS1      qc0       -8.6280302422805661e+00BOUNDS LO BND1      C0        -5 UP BND1      C0        5 LO BND1      C1        -5 UP BND1      C1        5QCMATRIX   qc0         C0        C0        9.9999999999994527e-01    C0        C1        5.0000000000000033e-01    C1        C0        5.0000000000000033e-01    C1        C1        5.0000000000000033e-01ENDATA

Many thanks!

Jack

Hi Jack,

The coefficients of your QCMatrix suffer from round off errors. The values should most likely read

QCMATRIX   qc0         C0        C0        1    C0        C1        0.5    C1        C0        0.5    C1        C1        0.5

These small round off errors may cause a lot of trouble during the solution process as they are far below the feasibility tolerance of the solver but still may have influence on the solution process. Moreover, it seems like the digits way below feasibility tolerance of the linear coefficients have significant meaning to the solution of the problem. This explains the sub-optimal solution when minimizing the modified problem. You can tackle this with setting NumericFocus=3 and BarCorrectors=100 to achieve additional solution stability.

If I write the MPS file to an LP file, which holds only limited precision of the coefficients, the barrier algorithm does not run in a sub-optimal solution. This confirms the influence of the round-off errors and digits below the feasibility tolerance in the coefficients.

For more information on numerics, I recommend the excellent article by Ed Klotz on Numerical Instability in Linear and Integer Programs. The article focuses on linear programs, but the main ideas also apply to quadratic programs.

Best regards,
Jaromił