Model convex or indefinite objectives with linear restrictions and limits. Review KKT-style candidates clearly now. Plot feasible regions, export summaries, and verify optimal points.
| # | x | y | Objective | Active Constraints | Generated From |
|---|---|---|---|---|---|
| 1 | 1.428571 | 2.285714 | -12.571429 | None | Interior stationary point |
| 2 | 1.428571 | 3.285714 | -11.571429 | C2 | Edge stationary point on C2 |
| 3 | 2.187500 | 2.437500 | -11.281250 | C3 | Edge stationary point on C3 |
| 4 | 2.000000 | 3.000000 | -11.000000 | C1, C2, C3 | Intersection: C1 & C2 | Intersection: C1 & C3 | Intersection: C2 & C3 | Edge endpoint on C1 | Edge endpoint on C2 | Edge endpoint on C3 |
| 5 | 0.000000 | 3.000000 | -9.000000 | x ≥ xmin | Edge stationary point on x = xmin |
| 6 | 0.000000 | 4.000000 | -8.000000 | C2, x ≥ xmin | Intersection: C2 & x = xmin | Edge endpoint on C2 | Edge endpoint on x = xmin |
| 7 | 2.000000 | 0.000000 | -8.000000 | y ≥ ymin | Edge stationary point on y = ymin |
| 8 | 3.000000 | 0.000000 | -6.000000 | C3, y ≥ ymin | Intersection: C3 & y = ymin | Edge endpoint on C3 | Edge endpoint on y = ymin |
| 9 | 0.000000 | 0.000000 | 0.000000 | x ≥ xmin, y ≥ ymin | Intersection: x = xmin & y = ymin | Edge endpoint on x = xmin | Edge endpoint on y = ymin |
The contour map shows the objective. The dot cloud marks feasible grid samples. The star marks the reported optimum.
| Item | Sample Value |
|---|---|
| Goal | Minimize |
| Objective | 0.5(4x² + 2xy + 2y²) - 8x - 6y |
| Constraint 1 | x + y ≤ 5 |
| Constraint 2 | x + 2y ≤ 8 |
| Constraint 3 | 3x + y ≤ 9 |
| Bounds | 0 ≤ x ≤ 6 and 0 ≤ y ≤ 6 |
| Expected Optimal Point | (1.428571, 2.285714) |
| Expected Objective | -12.571429 |
This calculator solves a two-variable quadratic program with bounded decision space. The objective is f(x,y) = 0.5(ax² + 2bxy + cy²) + dx + ey + f₀.
Linear restrictions use the form αx + βy ≤ rhs, αx + βy ≥ rhs, or αx + βy = rhs. The solver evaluates interior stationary points, boundary stationary points, and boundary intersections.
For stationary points, it solves the first-order system: ax + by + d = 0 and bx + cy + e = 0. It then filters every candidate through feasibility tests and ranks them by the chosen goal.
It optimizes a two-variable quadratic objective with optional linear inequalities, equalities, and finite bounds. You can solve either minimization or maximization cases within the chosen search region.
Bounds keep the search region finite and support stable contour plotting. They also help avoid ambiguous cases where a problem could extend without limit.
Yes. Choose the equals relation in any constraint row. The solver treats that line as mandatory and only keeps candidate points that satisfy it within a small tolerance.
The page checks interior stationary points, line-based stationary points, edge endpoints, and boundary intersections. Every feasible candidate is evaluated, then the best objective value is selected.
Active constraints are the restrictions that are tight at the reported optimum. In practice, their left and right sides match within numerical tolerance.
The contour surface is exact for the objective on the plotted grid. The feasible cloud is a sampled visualization, while the reported optimum comes from analytic candidate checks.
That usually means your bounds and constraints do not overlap, or one enabled constraint row is invalid. Adjust the relations, right sides, or bounds and solve again.
The determinant helps describe curvature, but it does not replace feasibility analysis. Use it together with the candidate table, active constraints, and plotted region.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.