Calculator Inputs
Objective form: f(x, y) = ax² + by² + cxy + dx + ey + f₀. Constraint form: px + qy = r.
Example Data Table
Use this sample set to test the calculator and compare your output.
| Item | Example Value | Description |
|---|---|---|
| Mode | MIN | Optimization target |
| a | 2 | Coefficient of x² |
| b | 1 | Coefficient of y² |
| c | 1 | Coefficient of xy |
| d | -8 | Coefficient of x |
| e | -6 | Coefficient of y |
| f₀ | 0 | Constant term |
| p | 1 | Constraint weight on x |
| q | 1 | Constraint weight on y |
| r | 6 | Constraint right side |
Expected Example Output
Formula Used
[ c 2b q ] [ y ] = [ -e ]
[ p q 0 ] [ λ ] [ r ]
The calculator solves the stationary point of a quadratic objective under one linear equality constraint. It then checks the constrained second-order behavior using the feasible direction vector (q, -p).
If κ is positive, the stationary point is a constrained minimum. If κ is negative, it is a constrained maximum. A near-zero κ means the curvature test is inconclusive.
How to Use This Calculator
- Choose whether you want to minimize or maximize the objective.
- Enter the quadratic objective coefficients a, b, c, d, e, and f₀.
- Enter the equality constraint coefficients p, q, and r for px + qy = r.
- Press Solve Optimization to compute x*, y*, λ*, and the objective value.
- Review the residual to confirm feasibility and the curvature value to classify the stationary point.
- Use the CSV and PDF buttons to export the current result summary.
- Use the chart to inspect the constraint line against the objective contours visually.
Frequently Asked Questions
1) What does this calculator optimize?
It solves a quadratic objective in x and y under one linear equality constraint. The output includes the stationary point, multiplier, objective value, feasibility residual, and constrained curvature test.
2) What happens if the system determinant is zero?
A singular system means the stationary point is not uniquely determined. This often happens with degenerate coefficients, overlapping conditions, or infinitely many feasible stationary candidates.
3) Does minimize or maximize change the computed x and y?
For this equality-constraint setup, the stationary equations use the same convention. The selected mode mainly helps interpret whether the stationary point fits your optimization target.
4) What does the Lagrange multiplier mean here?
The multiplier measures local sensitivity to the constraint. A larger absolute value suggests the constraint has a stronger shadow-price effect near the solved point.
5) Why is reduced curvature important?
It tests the objective only along the feasible direction. Positive curvature indicates a constrained minimum, negative curvature indicates a constrained maximum, and a near-zero value is inconclusive.
6) Can I enter decimal and negative values?
Yes. Every numeric field accepts decimals and negatives, which makes the tool suitable for shifted quadratics, mixed terms, and scaled equality constraints.
7) Why does the calculator show a residual?
The residual verifies numerical feasibility after solving. A value very close to zero confirms the returned point satisfies the equality constraint within floating-point tolerance.
8) Does this version support multiple constraints?
No. This version handles two variables with one equality constraint. For multiple constraints or nonlinear constraints, use a more general symbolic or numerical optimization method.