Difficulties during Optimization

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Difficulties during Optimization

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Numerical difficulties can degrade performance of the ILOG CPLEX Barrier Optimizer or even prevent convergence toward a solution. There are several possible sources of numerical difficulties:

elimination of too many dense columns may cause numerical instability;
tight convergence tolerance may aggravate small numerical inconsistencies in a problem;
unbounded optimal faces may remain undetected and thus prevent convergence.

The following subsections offer guidance about overcoming those difficulties.

Numerical Instability Due to Elimination of Too Many Dense Columns

Detecting and Eliminating Dense Columns explains how to change parameters to encourage ILOG CPLEX to detect and eliminate as many dense columns as possible. However, in some problems, if ILOG CPLEX removes too many dense columns, it may cause numerical instability.

You can check how many dense columns ILOG CPLEX removes by looking at the preprocessing statistics at the beginning of the log file, as explained in Preprocessing in the Log File. If you observe that the removal of too many dense columns results in numerical instability in your problem, then increase the column nonzeros parameter, BarColNz.

The default value of the column nonzeros parameter is 0 (zero); that value tells ILOG CPLEX to calculate the parameter automatically.

To see the current value of the column nonzeros parameter - either one you have set or one ILOG CPLEX has automatically calculated - you need to look at the level two display, by setting the BarDisplay parameter to 2.

If you determine that the current value of the column nonzeros parameter is inappropriate for your problem and thus tells ILOG CPLEX to remove too many dense columns, then you can increase the parameter BarColNz to keep the number of dense columns removed low.

Small Numerical Inconsistencies and Tight Convergence Tolerance

If your problem contains small numerical inconsistencies, it may be difficult for the ILOG CPLEX Barrier Optimizer to achieve a satisfactory solution at the default setting of the complementarity convergence tolerance. In such a case, you should increase the convergence tolerance parameter (BarEpComp for LP or QP models, BarQCPEpComp for QCP models).

Unbounded Variables and Unbounded Optimal Faces

An unbounded optimal face occurs in a model that contains a sequence of optimal solutions, all with the same value for the objective function and unbounded variable values. The ILOG CPLEX Barrier Optimizer will fail to terminate normally if an undetected unbounded optimal face exists.

Normally, the ILOG CPLEX Barrier Optimizer uses its barrier growth parameter, BarGrowth, to detect such conditions. If this parameter is increased beyond its default value, the ILOG CPLEX Barrier Optimizer will be less likely to determine that the problem has an unbounded optimal face and more likely to encounter numerical difficulties.

Consequently, you should change the barrier growth parameter only if you find that the ILOG CPLEX Barrier Optimizer is terminating its work before it finds the true optimum because it has falsely detected an unbounded face.

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