Introduction
Linear programming (LP) is a go-to method for solving optimization problems in everything from supply chains to finance. But even the most elegant LP models can run into trouble—not because of logic errors, but due to numerical difficulties that creep in during computation.
Let’s explore what these issues are, why they happen, and how you can avoid them.
What Are Numerical Difficulties?
Numerical difficulties arise when the math behind your LP model becomes unstable or inaccurate due to the way computers handle numbers—especially floating-point arithmetic. These issues can lead to:
- Slow performance
- Incorrect solutions
- Solver crashes
- Misleading feasibility or optimality reports
Common Culprits (with Real-World Examples)
1. Ill-Conditioned Matrices
When your constraint matrix has values that vary wildly in magnitude, solvers can struggle.
Example: A harvest scheduling model with volumes in board-feet and individual areas smaller than 1/10th acre might include coefficients ranging from 10−6 to 106 (or worse! I've seen it). This can cause the solver to miscalculate dual values or even fail to converge.
Harvest scheduling models typically represent large areas of land (thousands or even millions of acres). Stand volumes are similarly often hundreds of units per acre, along with prices that are also usually hundreds of dollars per unit. Multiplying these things together, it is possible to have objective function values in the hundreds of millions of dollars (greatly exceeding the
10−4 to 104 range that many solvers recommend). If your unscaled objective function is in the range of 100 million to 1 billion, you can always divide the outputs used in the objective function by 1,000,000 to get coefficients within the desired range.2. Floating-Point Precision Errors
Computers can’t represent every number exactly, especially decimals. This leads to rounding errors.
Example: A constraint like x+y=1 might yield x=0.999999999, y=0.000000001. Technically correct, but practically problematic—especially if your application requires clean, interpretable results. As a result, you should typically carry as many decimal places as possible, particularly in coefficients like area. For example, instead of an area record with 504.1, use 504.09775354.
3. Degeneracy
Degeneracy occurs when multiple basic feasible solutions exist at the same vertex. This can slow down or confuse the solver.
Example: In a network flow model (i.e., roads), several paths might have the same cost. The simplex method could cycle or take many iterations to find the optimal solution. Reprocess the network to eliminate redundancies by applying a shortest path algorithm.
4. Scaling Issues
When variables or constraints differ by several orders of magnitude, solvers may make poor pivoting decisions.
Example: One constraint is 0.0001x+0.0002y≤1, another is 1000x+2000y≤5000. The solver might misinterpret the importance of each constraint due to the scale difference. A simple fix is to manually scale the constraints to be within the desired range
(10−4 to 104) by multiplying both sides of the inequality: x+ y≤10000, x+ y≤5.5. Infeasibility and Unboundedness Confusion
Sometimes, numerical noise makes it hard to tell if a model is truly infeasible or unbounded.
Example: Nearly parallel constraints might appear to intersect due to rounding, leading the solver to falsely report feasibility. These ones may be difficult to spot. As always, if you start experiencing infeasibility or difficulties solving, consider backtracking by removing constraints and then adding them back in one at a time to see which one(s) cause(s) problems.
How to Fix or Avoid These Issues
Here are some practical tips:
- Normalize Your Model: Keep coefficients within a similar range. Avoid extreme values.
- Validate Results: After solving, double-check constraint satisfaction and objective values—especially in critical applications.
- Use Solver Preprocessing: Most solvers offer automatic scaling and presolve routines. However, they can't necessarily fix every scaling problem.
- Adjust Tolerances: Unless you know what you're doing, you are better off leaving default tolerances alone. However, sometimes an adjustment can fine-tune feasibility and optimality tolerances to reduce rounding errors. Consult your server's online help and documentation.
Final Thoughts
Numerical difficulties in LP models are often overlooked but can have serious consequences. By understanding the root causes and applying smart modeling practices, you can build more robust, reliable optimization models.
Have you run into any of these issues in your own work? Let’s talk about it in the comments!
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