Calculator
Example Data Table
| Objective | Goal | Bounds | Constraints | Step |
|---|---|---|---|---|
| x^2 + y^2 + sin(x*y) | Minimize | -3 ≤ x ≤ 3, -3 ≤ y ≤ 3 | x + y ≤ 4, x ≥ -2, y ≥ -2 | 0.5 |
| 10*x - x^2 + 6*y - y^2 | Maximize | 0 ≤ x ≤ 6, 0 ≤ y ≤ 6 | x + y ≤ 8 | 0.25 |
| exp(-x^2-y^2) + x*y | Maximize | -2 ≤ x ≤ 2, -2 ≤ y ≤ 2 | x^2 + y^2 ≤ 3 | 0.2 |
Formula Used
The calculator evaluates a nonlinear objective function over a bounded grid.
Objective: optimize f(x,y).
Feasibility: each point must satisfy every constraint within tolerance.
Minimum rule: choose the feasible point with the smallest f(x,y).
Maximum rule: choose the feasible point with the largest f(x,y).
Grid rule: x and y move from their lower bounds to upper bounds by the selected step size.
This method is practical for study, testing, and small two-variable models. Smaller step sizes usually improve accuracy. They also increase calculation time.
How to Use This Calculator
Enter the objective function using x and y. Use operators like +, -, *, /, and ^. You may also use functions such as sin, cos, sqrt, log, exp, pow, min, and max.
Select minimize or maximize. Enter lower and upper bounds for x and y. Add one constraint per line. Choose a step size. Use a smaller step for finer search. Press Calculate. Review the best feasible point and ranked candidates.
Nonlinear Programming Guide
What Nonlinear Programming Means
Nonlinear programming studies optimization models where the objective or constraints are not straight lines. The model may include powers, products, roots, logarithms, or trigonometric terms. These features make the problem more realistic. They also make the solution harder than linear programming.
Why This Calculator Helps
This calculator gives a practical search method for two-variable nonlinear problems. It checks many points inside the selected bounds. Then it tests every point against the entered constraints. Feasible points are ranked by the selected goal. The best point is shown first, with its objective value.
Using Bounds Correctly
Bounds are important in nonlinear programming. They define the search region. Wide bounds may scan many points. Narrow bounds focus the search. A good first run often uses broad bounds and a larger step. A second run can use tighter bounds around the best point.
Choosing the Step Size
The step size controls precision. A large step is fast, but it may miss a better nearby point. A small step gives a finer search, but it needs more calculations. For many classroom problems, start with 0.5 or 0.25. Then reduce it after seeing the first result.
Understanding Constraints
Constraints limit the allowed values of x and y. They can represent budget limits, area limits, production limits, or design restrictions. This calculator supports equality and inequality constraints. A tolerance is used because decimal calculations are rarely exact.
Reading the Result
The best point is the strongest feasible candidate found by the grid. The top table shows other good candidates. This helps you compare nearby points. It also helps you understand whether the solution is stable or sensitive to small changes.
Good Modeling Practice
Always check the formula before trusting the result. Use parentheses when needed. Compare the answer with a manual estimate. If the result seems strange, review the constraints, bounds, and step size. Nonlinear models can have several local best points.
FAQs
What is nonlinear programming?
Nonlinear programming is optimization with a nonlinear objective or nonlinear constraints. The formula may include powers, products, roots, logs, or trigonometric terms. It is common in engineering, finance, economics, and applied mathematics.
Does this calculator solve every nonlinear problem exactly?
No. It uses a grid search over selected bounds. It gives the best feasible point found on that grid. Smaller step sizes can improve accuracy, but they increase calculation time.
Which variables can I use?
This version supports two variables: x and y. You can enter formulas using both variables. You can also write constraints involving x, y, constants, and supported math functions.
How should I write constraints?
Write one constraint per line. Valid examples include x + y <= 5, x >= 0, y < 4, and x^2 + y^2 <= 9.
What step size should I choose?
Use a larger step for a quick scan. Use a smaller step for finer results. A good workflow is to start broad, then narrow the bounds near the best candidate.
Why is tolerance included?
Tolerance allows small numerical differences when checking constraints. This is useful for equality constraints and decimal calculations. A smaller tolerance is stricter. A larger tolerance is more forgiving.
Can I download the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple printable report. Both downloads are generated from the current form values.
Why did I get no feasible point?
Your constraints may conflict, or the feasible region may sit between grid points. Try wider bounds, a smaller step size, or check each constraint for typing errors.