Formula used
Minimize or maximize: ½xᵀQx + cᵀx + d
Subject to: Aineq x ≤ bineq
Aeq x = beq
lower ≤ x ≤ upper
The matrix Q controls curvature, while vector c controls the linear slope. Linear constraints define the feasible set available to the optimizer. Bounds restrict individual variables without adding separate constraint rows.
How to use this calculator
- Select minimization or maximization and choose the variable count.
- Enter the square Q matrix, linear vector c, and optional constant d.
- Add equality or inequality constraints and define variable bounds.
- Enter a reasonable starting point, then select numerical tolerances.
- Press the solve button and inspect feasibility, residuals, and KKT diagnostics.
- Export the result as CSV or print the page as a PDF report.
Worked example data
| Element | Example value | Meaning |
|---|---|---|
| Direction | Minimize | Search for the smallest objective value. |
| Q | [[2, 0], [0, 2]] | Creates a convex bowl-shaped objective. |
| c | [-2, -5] | Moves the unconstrained minimum. |
| Constraint | x₁ + x₂ ≤ 2 | Limits the combined variable total. |
| Bounds | x₁, x₂ ≥ 0 | Requires nonnegative decisions. |
Understanding quadratic programming
Quadratic programming optimizes a quadratic objective over linear restrictions. It appears in portfolio design, control systems, allocation, fitting, and production planning. Convex models provide the strongest global optimality guarantees for users.
A symmetric positive semidefinite Q matrix creates convex minimization. Positive definiteness usually produces one unique optimum when feasible. Negative eigenvalues warn that multiple stationary points may exist.
Linear equalities force solutions onto lines, planes, or hyperplanes. Inequalities carve out one side of each linear boundary. Variable bounds efficiently represent capacities, limits, and nonnegative quantities directly.
The calculator applies projected numerical steps with growing constraint penalties. Backtracking reduces unstable movements when the objective rises unexpectedly. Smooth and active-set inspired modes alter constraint treatment during iterations.
Residuals show how closely each reported solution satisfies restrictions. An active constraint lies close to equality at the solution. Large violations indicate infeasibility, weak convergence, or unsuitable numerical settings.
KKT conditions combine feasibility, stationarity, multipliers, and complementary slackness. For convex problems, satisfied KKT conditions strongly support global optimality. Approximate multipliers here are estimated from the detected active set.
Scaling matters because huge coefficient differences can harm convergence. Normalize variables or equations when magnitudes differ by many orders. Better starting values can also reduce penalty stages and iterations.
Non-convex problems require caution because local points may appear optimal. Try several starting points and compare objective values carefully. Industrial decisions should be verified with specialized optimization libraries independently.
Frequently asked questions
What is the Q matrix?
Q contains the quadratic coefficients and determines objective curvature. It should normally be symmetric for standard quadratic programming.
Why does the formula include one-half?
The one-half factor makes the gradient equal Qx plus c when Q is symmetric. It simplifies derivative calculations without changing the optimizer after coefficients are defined consistently.
What makes a quadratic program convex?
A minimization problem is convex when Q is positive semidefinite and constraints are linear. The calculator estimates this property from Q eigenvalues.
Can this calculator maximize a quadratic function?
Yes. Maximization is internally converted into minimization by changing the objective sign before numerical optimization.
What does an active constraint mean?
An active constraint is approximately equal at the reported solution. Such constraints often determine the final optimum location.
Why is no feasible solution confirmed?
Constraints may contradict each other, bounds may conflict, or iterations may be insufficient. Check residuals and try improved scaling or settings.
Are the Lagrange multipliers exact?
No. They are approximate least-squares estimates based on the detected active set and final objective gradient.
Can it solve non-convex problems?
It can search for a feasible stationary point, but global optimality is not guaranteed. Multiple starting points should be tested.
Is this suitable for critical engineering or financial decisions?
Use it for learning, checking, and preliminary analysis. Critical decisions should be verified with audited data and professional optimization software.
Numerical limitations
This single-file calculator uses dependency-free penalty methods rather than a commercial QP library. Results can depend on coefficient scaling, initial values, selected tolerances, and convexity. Always verify important solutions independently before applying them to real systems.