Analyze objective values, slack variables, and search directions. Review iteration history with practical optimization insights. Build reliable answers from constraints, tolerances, and diagnostics quickly.
| Item | Value | Meaning |
|---|---|---|
| Goal | Minimize | Searches for the lowest feasible objective value. |
| Objective | 0.5(2x² + 2·0.5xy + 2y²) - 6x - 8y | A convex quadratic with a cross term. |
| Initial point | (1, 1) | Strictly feasible starting point for barrier iterations. |
| Constraints | x + y ≤ 6, x ≥ 0, y ≥ 0, x ≤ 4, y ≤ 5 | Defines the feasible polygon used by the solver. |
| Barrier settings | t0 = 1, mu = 8, tolerance = 1e-6 | Controls accuracy, central path updates, and stopping. |
The calculator solves a two variable quadratic program with linear inequality constraints. The original model is:
Minimize or maximize f(x, y) = 0.5(q11x² + 2q12xy + q22y²) + c1x + c2y + k
Subject to ai x + bi y ≤ ci for each constraint row.
The interior point method replaces the hard boundary with a logarithmic barrier. For minimization, the merit function is:
Φ(x, y; t) = t f(x, y) - Σ log(ci - ai x - bi y)
The gradient combines the scaled objective gradient and the barrier gradient. The Hessian combines the quadratic Hessian and the barrier curvature terms. Each Newton direction solves HΔ = -g. A backtracking line search keeps every slack positive and reduces the barrier merit function.
The solver updates the barrier parameter by t ← μt. The estimated duality gap is m / t, where m is the number of inequalities. Convergence is strongest when the transformed quadratic term is convex and the initial point is strictly feasible.
An interior point method searches the feasible region from the inside. It avoids stepping onto a hard boundary by attaching a logarithmic barrier to every inequality. That strategy is useful when you want a smooth numerical path, detailed diagnostics, and a repeatable stopping rule.
This calculator focuses on a practical two variable quadratic model. Many teaching examples in optimization use exactly this form because it lets you inspect curvature, gradients, and feasibility without hiding the mathematics. The output is numerical, so it helps students, analysts, and educators compare inputs and test solver behavior quickly.
The iteration table is important because a final answer alone rarely explains convergence quality. A good optimization report also shows the barrier parameter, gradient norm, step size, merit value, and slack size at each stage. Those fields reveal whether the method is stabilizing, stalling, or moving toward a tighter central path.
Strict feasibility matters here. If the initial point violates even one inequality, the logarithm becomes undefined and the barrier model breaks. That is why the tool checks all slacks before it starts. A feasible point with positive slack gives the Newton system room to move safely.
The graph gives a visual view of objective movement across iterations. When the problem is well posed, the sequence usually settles and the gradient norm shrinks. If your quadratic term is not convex after the chosen goal transformation, you may still get numbers, but they should be interpreted with caution.
Use the example dataset to test the full workflow first. Then replace the coefficients and constraints with your own model, compare iteration logs, export the table, and keep a clean report for review or teaching.
It handles two variable quadratic objectives with linear inequality constraints. The method is best for convex minimization or concave maximization after sign transformation.
The barrier term uses logarithms of slack values. A zero or negative slack makes the logarithm undefined, so the algorithm cannot start safely.
It gradually shifts emphasis from the barrier to the original objective. Larger t values usually move the solution closer to the constrained optimum.
For this barrier model, the estimate is m divided by t, where m is the number of inequalities. Smaller values indicate a tighter approximation.
Failure can happen from poor feasibility, singular Hessians, nonconvex transformed models, or settings that force very small line search steps.
Write one inequality per line using three numbers: a b c. The calculator interprets that line as a*x + b*y <= c.
The Plotly graph shows the objective value by iteration. It helps you see whether the numerical path is stabilizing or oscillating.
You can export the iteration history as CSV and generate a PDF report from the result section for review or sharing.
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.