Linear Programming Part 9- Understanding Solver Output<

Reading solver status and validating optimization results. Solver status, solution validation, common solver outcomes, practical interpretation

1. Introduction

An optimization solver does not always return a usable optimal solution.

Depending on the LP formulation and the solver's execution, the result may be:

An optimal solution
No feasible solution
An unbounded solution
An incomplete solve due to computational limits

Therefore, getting output from a solver is not the same as having a valid optimization result.

Before using the solution, we should check:

\[ \boxed{ \text{Solver Status} \rightarrow \text{Solution} \rightarrow \text{Validation} } \]

This part focuses on reading solver output and checking whether the result can be safely used.

2. Solver Status

The solver status tells us what happened when the solver finished.

A solver may return output even when an optimal solution was not found. Therefore, we should check the status before using the result.

Common solver outcomes are:

Status	Meaning
Optimal	A valid optimal solution was found
Infeasible	No solution satisfies all constraints
Unbounded	The objective can improve without a finite limit
Time/Iteration Limit	The solver stopped before proving optimality
Numerical Difficulty	The solver encountered numerical problems

For example, with SciPy:

result = linprog(
    c=[-10, -15],
    A_ub=[[2, 4], [3, 2]],
    b_ub=[20, 18],
    bounds=[(0, None), (0, None)],
    method="highs"
)

print(result.success)
print(result.status)
print(result.message)

The main fields to check are:

result.success: whether the solver successfully found an optimal solution
result.status: numerical code describing the solver outcome
result.message: explanation of the outcome

For example:

True
0
Optimization terminated successfully.

This indicates that the solver successfully found an optimal solution.

If the status indicates infeasible, unbounded, or a computational problem, the result should not be treated as a confirmed optimal solution.

Key point: Always check the solver status before using optimization results.

3. Solution Validation

After the solver reports a successful result, we should validate the solution before using it.

The main checks are:

Decision variables are valid
All constraints are satisfied
Objective value matches the reported solution

For the LP from earlier, suppose the solver returns:

result.x

with:

[4. 3.]

This means:

\[ x = 4,\qquad y = 3 \]

We can check the constraints manually:

\[ 2(4)+4(3)=20 \]

\[ 3(4)+2(3)=18 \]

Both constraints are satisfied.

We can also verify the objective:

\[ 10(4)+15(3)=85 \]

So the reported solution is internally consistent.

For practical applications, validation is important because solver output can be affected by:

Incorrect model formulation
Incorrect input data
Numerical precision
Unexpected solver termination

Key point: A successful solver status is not a substitute for checking that the returned solution actually satisfies the model.

4. Common Solver Outcomes

Different solver outcomes require different actions.

Infeasible

The constraints cannot all be satisfied simultaneously.

For example:

\[ x \geq 10 \]

and

\[ x \leq 5 \]

cannot both be true.

What to check:

Conflicting constraints
Incorrect constraint values
Missing or incorrect variable bounds
Overly restrictive requirements

Unbounded

The objective can continue improving without reaching a finite optimum.

For example, a maximization problem might allow:

\[ x \rightarrow \infty \]

without any constraint limiting \(x\).

What to check:

Missing constraints
Incorrect constraint directions
Missing variable bounds

Time or Iteration Limit

The solver stopped before completing the optimization.

This does not necessarily mean the problem is infeasible or unbounded. It means the solver did not finish within the allowed computational limit.

What to check:

Solver runtime
Problem size
Number of variables and constraints
Whether the formulation can be simplified

Numerical Difficulty

The solver encountered numerical problems while processing the model.

What to check:

Very large or very small coefficients
Poorly scaled data
Nearly redundant constraints
Numerical precision issues

Key point: The solver outcome tells us what happened. It also tells us what should be checked before the result is used.

5. Practical Interpretation

Once the solver result has been validated, the next step is to translate the output into a decision that can be used in practice.

A solver may return values such as:

result.x

which could give:

[4. 3.]

These values only become useful when mapped back to the original decision variables:

\[ x = 4 \quad \text{Product A} \]

\[ y = 3 \quad \text{Product B} \]

The objective value tells us the resulting performance:

\[ \text{Profit} = 85 \]

A practical interpretation therefore connects:

\[ \boxed{ \text{Solver Output} \rightarrow \text{Decision Variables} \rightarrow \text{Business Meaning} } \]

Before using the result, confirm:

What does each variable represent?
What does the objective value represent?
Are all constraints satisfied?
Is the solution actually implementable?
Are the assumptions and input values still valid?

This final step is important because an optimization solver solves the mathematical model, not the real-world problem itself.

Key point: Solver output becomes useful only after it is interpreted in the context of the original problem.

6. Summary

Before using an LP solver result, follow a simple validation process:

\[ \boxed{ \text{Solver Status} \rightarrow \text{Solution Validation} \rightarrow \text{Practical Interpretation} } \]

Solver Status tells us what happened during optimization.
Solution Validation confirms that the returned values satisfy the model.
Practical Interpretation maps the mathematical result back to the real-world decision.

The main lesson is:

A solver returning a result does not automatically mean the result is a valid, usable optimal solution.

Always check the status, validate the solution, and then interpret it in the context of the original problem.

Relevant Link(s)

AI in Context Main Page