Convex Optimization Tool

Model quadratic objectives, inspect Hessians, and compare updates. Handle box limits with projected search automatically. Turn optimization inputs into practical decisions, tables, and exports.

Enter Optimization Inputs

This single-page tool solves two-variable quadratic convex programs with optional box constraints.

Example Data Table

q11 q12 q22 c1 c2 Bounds Start Approx x* Approx minimum
4 1 3 -8 -6 x1, x2 in [0, 5] (0, 0) (1.6364, 1.4545) -10.9091

Formula Used

Objective: f(x1, x2) = 0.5(q11x12 + 2q12x1x2 + q22x22) + c1x1 + c2x2 + k

Gradient: ∇f(x) = [q11x1 + q12x2 + c1, q12x1 + q22x2 + c2]

Hessian: H = [[q11, q12], [q12, q22]]

Convexity rule: the model is convex when the Hessian is positive semidefinite. For this 2×2 symmetric case, the smallest eigenvalue must be nonnegative.

Projected update: x(k+1) = Π[l,u](x(k) − α∇f(x(k)))

Stopping check: the solver stops when both the step norm and projected gradient mapping norm drop below tolerance.

How to Use This Calculator

  1. Enter the symmetric Hessian entries q11, q12, and q22 for your quadratic model.
  2. Add the linear coefficients c1 and c2, plus any constant term k.
  3. Choose starting values, box bounds, solver method, learning rate, tolerance, and iteration limit.
  4. Submit the form to display the final point, minimum value, eigenvalue diagnostics, and iteration table above the form.
  5. Download CSV for raw history or PDF for a shareable report.
  6. If the convexity warning appears, revise the Hessian to keep the optimization model convex.

FAQs

1. What kind of problem does this tool solve?

It solves two-variable quadratic optimization models with optional lower and upper bounds. The objective is evaluated with gradient-based iterative updates and convexity checks.

2. How does the tool decide whether the model is convex?

It computes the Hessian eigenvalues. When the smallest eigenvalue is nonnegative, the quadratic model is convex. A positive smallest eigenvalue means strong convexity.

3. Why are box bounds useful?

Bounds keep the solution inside allowed ranges. This is useful for resource limits, nonnegative variables, and practical decision constraints.

4. What does the projected gradient mapping norm mean?

It measures how close the current point is to satisfying first-order optimality under bounds. Smaller values indicate a better constrained solution.

5. When should I use Nesterov acceleration?

Use it when you want faster progress on smooth convex models. If updates oscillate, reduce momentum or learning rate.

6. What is a good learning rate?

A practical starting point is about 1 divided by the largest Hessian eigenvalue. Smaller values improve stability but may slow convergence.

7. Why does the stationary reference differ from the final bounded answer?

The stationary reference ignores bounds. If that point lies outside the allowed box, the constrained optimum moves to a feasible boundary or interior point.

8. Can I use this for many variables?

This page is designed for two variables to keep inputs readable and outputs interpretable. The same concepts extend to larger numerical solvers.

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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.