Solver Error: “Linearity conditions required by this LP Solver are not satisfied.”

The Solver error "Linearity conditions required by this LP Solver are not satisfied" occurs when you run the Simplex LP engine on an optimization model that contains non-linear math. The Simplex algorithm is designed specifically for linear programming. It requires that every mathematical relationship between your decision variables, constraints, and objective cell forms a straight line. If any formula in the calculation path breaks linearity, the pre-solver test detects the discrepancy and halts before running.

Quick Answer

This error means your model uses non-linear formulas (such as multiplying decision variables together, dividing by a variable, or using =IF(), =MIN(), =MAX(), or =VLOOKUP()). To resolve it, rewrite non-linear functions into linear algebraic constraints, or switch the Solver engine from Simplex LP to GRG Nonlinear.

---

What “Linearity” Means in Excel Optimization

For a model to be linear, every formula linking the Changing Variable Cells to the Objective Cell and Constraints must satisfy two mathematical principles:

1. Proportionality: If you multiply a decision variable by a factor, its contribution to the objective and constraints must scale by that exact same factor. (e.g., $5x$ is linear; $x^2$ is non-linear).
2. Additivity: The total effect of the variables must be the sum of their individual contributions. (e.g., $3x + 2y$ is linear; $x \times y$ is non-linear).

Visualizing the Difference:

Linear Function (Simplex LP):             Non-Linear Function (Breaks Simplex):
Objective                                Objective
   ^                                        ^             * * * (Peak)
   |           /                            |           *       *
   |          /                             |          *         *
   |         /                              |         *           *
   |        /                               |        *             *
   +-----------------> Variable             +-----------------------> Variable
   Constant slope at all points.            Variable slope (curves, thresholds, steps).

When you click Solve, the Simplex LP engine runs a numerical linearity check. It tests an arbitrary step change in your variables. If the resulting change in the objective or constraints does not match the expected proportional slope, Solver halts and displays the linearity error.

---

The 5 Most Common Linearity Breakers in Excel

Review the formulas connecting your decision variables to your target cells. Look for these five frequent culprits:

Common Linearity Violations:
   ├── 1. Discontinuous Logic ─────> =IF(), =IFS(), =SWITCH(), =CHOOSE()
   ├── 2. Variable Interactions ───> DecisionVar_A * DecisionVar_B, or Constant / DecisionVar
   ├── 3. Threshold Functions ─────> =MIN(), =MAX(), =ABS(), =ROUND(), =INT()
   ├── 4. Lookup Tables ───────────> =VLOOKUP(), =XLOOKUP(), =INDEX(..., MATCH(...))
   └── 5. Exponents / Powers ──────> DecisionVar^2, =SQRT(DecisionVar)

1. Logic Statements (IF, IFS)

Using =IF(C4 > 100, C4 * 10, C4 * 12) creates a step change in your calculation slope. The Simplex engine cannot evaluate derivatives across conditional logic breaks.

2. Multiplying Two Decision Variables

Writing =B4 * B5 where both cells are Changing Variable Cells violates additivity. For example, in portfolio optimization, multiplying asset weight $w_i$ by asset weight $w_j$ to calculate portfolio variance is quadratic, not linear.

3. Threshold Functions (MIN, MAX, ABS)

Formulas like =MAX(0, Production_Volume - 500) create an “inflection point” or kink in the curve. At the threshold, the slope abruptly changes from 0 to 1, which breaks Simplex LP requirements.

4. Lookup Formulas

Functions like =VLOOKUP(Changing_Cell, Pricing_Table, 2, FALSE) are non-smooth and discontinuous. Solver cannot compute a linear rate of change through an index search.

---

Step-by-Step Troubleshooting Guide

Troubleshooting Path:
Trace formula dependencies from Objective cell back to Variables
   │
   ├── Did you multiply variables together or use curved formulas (e.g. x^2)?
   │       └── YES ──> Switch Solver engine to "GRG Nonlinear".
   │
   ├── Did you use =MIN(), =MAX(), or =ABS()?
   │       └── YES ──> Linearize formulas using auxiliary constraints.
   │
   └── Did you use =IF() or lookup functions?
           ├── Can you express logic as 0/1 binary variables? ──> Linearize model.
           └── Logic is too complex to convert? ───────────────> Switch to "Evolutionary" engine.

Step 1: Trace Precedents

1. Select your Objective Cell.
2. Go to the Formulas tab and click Trace Precedents.
3. Follow the arrows back to the Changing Variable cells.
4. Repeat this check for every cell referenced in your Constraints window.
5. Inspect every formula along those paths for non-linear functions.

Step 2: Linearize Thresholds (Fixing MIN and MAX)

You can often keep the speed and precision of Simplex LP by converting non-linear formulas into auxiliary linear constraints.

• Problem Formula: Cost = MAX(Regular_Cost, Overtime_Cost) where both costs depend on labor hours.
• Linear Solution:
  1. Create a new changing decision variable cell: Max_Cost_Variable.
  2. Set your objective formula to reference Max_Cost_Variable directly.
  3. Add two new linear constraints in the Solver dialog:
     • Max_Cost_Variable >= Regular_Cost
     • Max_Cost_Variable >= Overtime_Cost

Because the minimization objective naturally pushes Max_Cost_Variable down, it will settle on the larger of the two costs without ever calling the non-linear =MAX() function.

Step 3: Choose the Correct Alternative Solver Engine

If your model cannot be linearized algebraically:

1. Open the Solver Parameters dialog (Data > Solver).
2. Locate the Select a Solving Method dropdown at the bottom.
3. Choose the engine that matches your formula architecture:
   • GRG Nonlinear: Select this if your formulas contain smooth curves, exponents, quadratic variances, or divisions by variables (e.g., curves with no sudden vertical steps).
   • Evolutionary: Select this if your model relies on discontinuous logic like =IF(), =ROUND(), or table lookups that cannot be rewritten. The Evolutionary engine uses genetic algorithms to search for solutions without requiring linear relationships.

---

Diagnostic Summary for Solver Failures

• If Simplex LP returns the linearity error: A curved, variable-multiplied, or discontinuous formula exists in your model’s calculation chain.
• If switching to GRG Nonlinear stops on a flat zone: Review initial variable values. Nonlinear algorithms require realistic starting guesses to find local gradients.
• If Solver reports that no feasible solution exists: Review your constraint boundaries for conflicting requirements. See Solver Error: “Solver could not find a feasible solution.”.
• If Solver runs indefinitely on an Evolutionary model: The iteration ceiling may be unconstrained. See Solver Error: “The Maximum Iterations limit was reached.”.