Excel Solver: GRG Nonlinear vs. Simplex LP

Use Simplex LP when the objective and every constraint are linear in the changing cells; use GRG Nonlinear when the model contains smooth nonlinear relationships. If the model has jumps or other nonsmooth logic, neither may be the best choice: Excel Solver also offers Evolutionary.

What the two Solver methods mean

Excel Solver changes selected worksheet cells, called variable cells or decision variables, to optimize a target cell while meeting constraints. The solving method determines how Solver searches for values. It does not change the meaning of the worksheet formulas, and choosing a method cannot make a wrongly structured model correct.

Simplex LP is the method for linear programming models. In a linear model, each changing cell appears only to the first power, is not multiplied by another changing cell, and is combined with fixed coefficients. The objective and constraints can include sums of decision variables, but they cannot have curved relationships between those variables.

GRG stands for Generalized Reduced Gradient. GRG Nonlinear is intended for smooth nonlinear models: models whose formulas change continuously and have a usable gradient, or rate of change, across the relevant range. Examples include a squared decision variable, a product of decision variables, or a smooth formula involving a changing cell.

How the methods search

For a continuous linear model, Simplex LP searches the feasible region formed by the constraints. That region has a flat, polygon-like geometry in two dimensions and a higher-dimensional equivalent in larger models. The simplex algorithm moves between corner points, or vertices, seeking a better objective value. For a feasible, bounded linear program, an optimum exists at a vertex, which is why this method fits linear models so well.

GRG Nonlinear instead uses gradients to move through a nonlinear landscape. It adjusts variable values in directions that are expected to improve the objective while respecting constraints. Curves, hills, valleys, and ridges can make that landscape more complicated than a linear model’s feasible region. As a result, GRG can find a local optimum—a solution better than nearby alternatives—without proving that no better solution exists elsewhere.

The distinction matters most when formulas are genuinely nonlinear, not merely because a worksheet looks complicated. A long sum with fixed coefficients can still be linear. A short formula that multiplies two decision cells is nonlinear.

Recognizing a linear model

Consider a small production plan with two products. The decision cells represent the quantities to make, and the objective is total profit. If each product has a fixed profit per unit, total profit is a sum of each quantity multiplied by its fixed profit. If each product uses fixed amounts of labor and material per unit, resource limits are also sums of quantities multiplied by fixed amounts. This is a linear model, so Simplex LP is the natural method.

By contrast, suppose the profit per unit changes with the quantity produced, perhaps because of a smooth discount curve. Or suppose a model includes a cost term based on the product of two changing quantities. Those relationships are nonlinear, so GRG may be appropriate if they are smooth. A formula containing a variable in a denominator is generally nonlinear as well, and it can behave badly near zero.

Some worksheet functions create a less straightforward case. IF, MIN, MAX, and lookup formulas can introduce jumps, switches, or corners when their results depend on changing cells. A model with such logic is not necessarily a good fit for GRG’s smooth-gradient approach. Some relationships can be reformulated with additional variables and constraints, but that requires careful modeling rather than simply changing the method dropdown.

When integer decisions are involved

A linear model can still include discrete decisions. For example, a factory may have to produce whole units, or a project may be either selected or not selected. These are integer or binary restrictions on decision cells. The underlying objective and resource relationships may remain linear, even though the allowed solutions are no longer continuous.

For a linear model with integer or binary restrictions, start with Simplex LP. Solver can use integer optimization techniques such as branch and bound to search among discrete choices. The search can take longer than the continuous problem, especially as the number of integer decisions grows. Be aware that a rounded continuous answer is not automatically a valid or optimal integer answer: rounding may break constraints or miss a better combination.

If integer decisions are combined with nonlinear relationships, the problem is more difficult. The appropriate method depends on the model’s structure and Solver’s capabilities; do not assume that selecting GRG makes every nonlinear-integer model reliably solvable. Check the returned status and solution, and consider a specialized optimization tool for a large or important model.

How to choose in Excel

On Excel for Microsoft 365 for Windows, open the workbook and choose Data > Solver. If Solver is not shown, enable the Solver Add-in through Excel’s add-in settings, then return to the Data tab. In Solver Parameters, set the objective cell, choose Max, Min, or a target value, identify the changing variable cells, and add the model’s constraints.

Before choosing a method, inspect the formulas that feed the objective and constraints. Ask whether each relationship is linear in the changing cells, whether any variables are multiplied or used as exponents, and whether the formulas contain conditional switches. Then choose the method that matches the model:

Choose Simplex LP for linear objectives and constraints, including linear models with integer or binary restrictions.
Choose GRG Nonlinear for smooth nonlinear objectives or constraints, such as a curved cost relationship.
Consider Evolutionary for discontinuous or nonsmooth models that are difficult to handle with gradients, while expecting a potentially slower search.

Select the method in Solver’s Select a Solving Method control, then choose Solve. On Mac, Solver is also an Excel add-in, but availability and dialog layout can vary by Excel version; use the Solver command on the Data tab and look for its solving-method control. The instructions here describe desktop Excel, not a guarantee that every Solver dialog or option is identical across platforms.

Why starting values and constraints matter

Starting values are especially important for GRG. Because it follows local information, different starting points can lead to different local optima in a nonconvex model. If the result seems unexpectedly poor, try reasonable alternative starting values, check whether the variables have realistic bounds, and compare the objective values from the runs. This can provide evidence that a better solution exists, but it does not by itself prove global optimality.

Constraints also help define a meaningful search. If a quantity cannot be negative, enter a nonnegative restriction or use the relevant Solver option. If a denominator must stay away from zero, model an appropriate lower bound. Bounds can prevent unrealistic values and numerical trouble, but they should reflect the real problem rather than being chosen merely to force a preferred answer.

For a linear model, a Solver report or status message should be read alongside the worksheet. A message indicating an optimum is useful, but verify that the constraints are satisfied to an acceptable tolerance and that units and signs are correct. If Solver reports that no feasible solution exists, the constraints may conflict; if the objective can improve without limit, a missing bound may be the cause.

Common mistakes and misleading results

Choosing Simplex because a model has many cells: model size does not determine the method. Linearity does.
Assuming every formula with multiplication is nonlinear: multiplying a decision variable by a fixed coefficient is linear. Multiplying two changing variables is nonlinear.
Using GRG for a formula with abrupt switches: IF-based rules or lookup-driven changes may not provide the smooth behavior GRG expects.
Treating a local optimum as the best possible answer: GRG may stop at a nearby optimum in a nonlinear model. Try alternative starting values and review the model’s shape.
Rounding continuous answers to meet integer requirements: use integer constraints in Solver instead; rounded values may violate limits or reduce the objective unnecessarily.
Trusting a result without checking units: mixing hours with minutes, or dollars with thousands of dollars, can produce a mathematically valid but practically meaningless result.

One frequent misconception is that Simplex LP is only for equations that look like textbook algebra. A worksheet may contain many cells, labels, and totals while still representing a linear model. Conversely, a model with only a few cells can be nonlinear if it includes a variable-variable product, a power, or a changing denominator. Inspect the dependency of each formula on the changing cells rather than judging by worksheet appearance.

When to use Evolutionary or another tool

Excel Solver’s Evolutionary method is an alternative when a model’s formulas are discontinuous, nonsmooth, or otherwise difficult for gradient-based GRG. It searches using an evolutionary approach rather than relying on the same local gradient search. That flexibility can come at the cost of speed and precision, and results should be checked against constraints and, where possible, other candidate solutions.

For a very large model, a model with many integer decisions, or a problem where a proven global optimum is essential, the standard Solver add-in may not be sufficient. Commercial optimization add-ins and dedicated linear or nonlinear optimization software can offer different engines, limits, and reporting. Compare their capabilities and licensing before moving a workbook model; the right alternative depends on the problem size and the level of certainty required.

Frequently Asked Questions

Is GRG Nonlinear better than Simplex LP?

Neither is universally better. Simplex LP is generally the right choice for a linear model, while GRG Nonlinear fits smooth nonlinear relationships. Matching the method to the formulas is more important than choosing one method as a default winner.

Can Simplex LP solve a model with binary variables?

Yes, when the objective and constraints are linear, Simplex LP is the appropriate method for a model with binary or integer restrictions. Solver must search the discrete choices, so it may take longer than solving the same model without those restrictions.

Why does GRG return different answers from different starting values?

GRG can converge to different local optima in a nonlinear model. Starting values affect the path of the search. Try sensible alternative starts and compare results, but remember that finding the same answer repeatedly is not proof that it is the global optimum.

Can I use Simplex LP if my objective has a SUM or SUMPRODUCT?

Often, yes. SUM and SUMPRODUCT can express linear relationships when the changing cells are combined with fixed coefficients. If the formula multiplies two changing ranges together, however, the model may be nonlinear. Check what depends on the decision cells.

What should I choose if my model uses IF statements?

It depends on what the IF tests and returns. If changing cells trigger abrupt switches, the model may be nonsmooth, so GRG may struggle; Evolutionary may be worth trying. If the logic can be represented as a linear model with additional decision variables and constraints, Simplex LP may still be possible.

Try it: the Date Difference Calculator: days, months, years and weekdays between two dates, with the DATEDIF formulas.

More in How-to basics

Sources: Function behaviour and menu paths: Microsoft Support documentation for Excel. Method & sources

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