Advanced Constrained Optimization Solver Calculator

Analyze quadratic objectives with practical constraints and visual outputs. Compare feasible points, export findings, and interpret optimization decisions confidently.

Calculator Inputs

This page solves a two-variable quadratic objective with optional equality and inequality constraints plus variable bounds.

Objective coefficient a for x²

Objective coefficient b for y²

Objective coefficient c for xy

Objective coefficient d for x

Objective coefficient e for y

Constant term f

Enable equality constraint

Equality coefficient for x

Equality coefficient for y

Equality right side value

Enable inequality constraint

Inequality coefficient for x

Inequality coefficient for y

Inequality right side value

Goal

Minimum x

Maximum x

Minimum y

Maximum y

Reset Inputs

Plotly Graph

The chart displays feasible candidate points and highlights the chosen optimum.

Example Data Table

Parameter	Example Value	Description
a, b, c, d, e, f	2, 2, 0, -8, -6, 0	Quadratic objective coefficients.
Equality	x + y = 5	Resource balance or exact requirement.
Inequality	x - y ≤ 2	Capacity or policy limit.
Bounds	0 ≤ x ≤ 5, 0 ≤ y ≤ 5	Allowed decision range.
Goal	Minimize	Select minimum or maximum objective value.

Formula Used

Objective: f(x, y) = ax² + by² + cxy + dx + ey + f

Equality constraint: g₁x·x + g₁y·y = h₁

Inequality constraint: g₂x·x + g₂y·y ≤ h₂

Bounds: xmin ≤ x ≤ xmax and ymin ≤ y ≤ ymax

Stationarity for interior points: ∂f/∂x = 2ax + cy + d, ∂f/∂y = 2by + cx + e

Equality KKT system: ∇f(x, y) = λ∇g(x, y)

Boundary search: evaluate box corners, line intersections, and active-constraint stationary points

This implementation checks feasible stationary candidates, boundary intersections, and corner points. It then evaluates the objective at every feasible candidate and returns the best result for the selected goal.

How to Use This Calculator

Enter the quadratic objective coefficients for x², y², xy, x, y, and the constant term.
Enable the equality constraint when an exact balance must hold.
Enable the inequality constraint when a limit or ceiling must be respected.
Provide lower and upper bounds for both variables.
Select whether you want to minimize or maximize the objective.
Click Solve Optimization to display the best feasible point above the form.
Review the candidate table and chart to understand why the solution was chosen.
Use the CSV or PDF buttons to export the result summary.

Frequently Asked Questions

1. What kind of optimization problem does this page solve?

It solves two-variable quadratic objectives with optional equality, inequality, and bound constraints. This structure fits many classroom, engineering, planning, and operations examples.

2. Does it support both minimum and maximum objectives?

Yes. Choose either minimization or maximization from the goal menu. The page evaluates feasible candidates and returns the best objective value for your selected goal.

3. Why are bounds important in constrained optimization?

Bounds restrict decision variables to realistic ranges. They also prevent impractical solutions and help define a closed feasible region for reliable evaluation.

4. What happens if no feasible point exists?

The calculator shows an error message. This usually means the chosen equality, inequality, and bounds conflict with each other and leave no valid solution region.

5. Is the solver using a numerical search method?

This version uses analytical candidate generation for stationary points and boundary checks. It is designed for transparent, explainable results instead of black-box iteration.

6. Can I use decimal values?

Yes. All coefficient, constraint, and bound fields accept decimals. That makes the calculator suitable for many continuous optimization examples.

7. Why does the chart show only candidate points?

The chart focuses on feasible evaluated candidates, which are the decisive points for the final answer. This keeps the visualization readable and useful.

8. When should I use an equality constraint?

Use it when the total must match an exact target, such as a budget split, material balance, or fixed allocation requirement.