Analyze constrained minima with bounds, penalties, steps, and refinements. Review iteration metrics with concise summaries. Build intuition with feasible checks, gradients, and exports included.
This example minimizes x² + y² - 6x - 4y under three constraints.
| Item | Value | Meaning |
|---|---|---|
| Objective coefficients | x²=1, y²=1, xy=0, x=-6, y=-4, c=0 | Creates a convex bowl with an unconstrained minimum near (3,2). |
| Constraint 1 | x + y - 5 ≤ 0 | Keeps the solution on or below the line x+y=5. |
| Constraint 2 | x ≥ 0 | Prevents negative x values. |
| Constraint 3 | y ≥ 0 | Prevents negative y values. |
| Bounds | x:[0,6], y:[0,6] | Restricts the search region. |
| Penalty settings | Start=2, Multiplier=4, Outer=6, Step=0.25 | Raises punishment for violations each outer iteration. |
| Expected optimum | Approximately (3,2) | The boundary point satisfies x+y=5 and minimizes the objective. |
Objective model: f(x,y) = ax² + by² + cxy + dx + ey + f₀
Constraint model: each constraint uses g(x,y) = px² + qy² + rxy + sx + ty + u
Inequality penalty: for g(x,y) ≤ 0, violation is max(0, g)
Reverse inequality penalty: for g(x,y) ≥ 0, violation is max(0, -g)
Equality penalty: for g(x,y) = 0, violation is |g|
Penalty function: Φ(x,y) = f(x,y) + r Σ(violation²)
Search method: the calculator performs a bounded grid scan, then refines the best point with a local pattern search.
Iteration logic: after each outer iteration, the penalty weight grows by the selected multiplier, pushing the solution toward feasibility.
It minimizes a two-variable quadratic objective under polynomial constraints. The solver converts violations into penalty terms and searches numerically inside your chosen bounds.
They turn constrained problems into easier unconstrained ones. By increasing penalty weights, infeasible points become less attractive, so the search drifts toward valid solutions.
Yes. Set the relation to equality. The calculator penalizes the absolute constraint value, which encourages the final point to land very close to zero.
A smaller step can improve accuracy, but it also increases computation. The calculator may enlarge the step automatically if the search grid becomes excessively large.
Check the maximum violation and the feasibility status. If the maximum violation is below your tolerance, the result is treated as feasible within that numerical threshold.
It estimates how flat the penalized surface is near the reported point. Smaller values usually suggest the algorithm reached a stronger stationary location.
No. It is a numerical search tool. It is useful for exploration, diagnostics, and bounded optimization, but it does not return symbolic proofs.
Well-scaled coefficients, realistic bounds, and sensible step sizes usually produce clearer results. Start with moderate penalties, then increase them if feasibility remains weak.
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.