Explore constrained variational inequality solving with affine operators. Inspect residuals, feasibility, and convergence behaviour instantly. Helpful for optimisation studies, teaching, and numerical decision work.
This calculator solves an affine variational inequality on a box-constrained feasible set. Enter matrix A, vector b, lower and upper bounds, an initial point, and iteration settings. The page then applies projected iterations and reports convergence, feasibility, solution values, and an iteration history table.
Use rows separated by new lines or semicolons. Separate numbers with spaces or commas. The page layout remains single-column, while the input grid becomes 3 columns on large screens, 2 on medium screens, and 1 on mobile.
| Parameter | Example Entry | Meaning |
|---|---|---|
| Matrix A | 2 -1 -1 2 |
Affine operator matrix for F(x) = Ax + b. |
| Vector b | -1 -2 | Constant shift in the operator. |
| Lower Bounds | 0 0 | Smallest allowed value in each component. |
| Upper Bounds | 5 5 | Largest allowed value in each component. |
| Initial Vector x0 | 0.5 0.5 | Starting point before projected iterations begin. |
| Step Size α | 0.25 | Controls the move length before projection. |
| Tolerance | 0.000001 | Required residual threshold for convergence. |
| Maximum Iterations | 200 | Safety cap for the iterative process. |
| Expected Behaviour | x* ≈ [1.333332, 1.666665] | Illustrative converged solution for the sample data. |
The calculator solves the affine variational inequality problem:
The projected iteration applied here is:
The stopping metric is the scaled projected residual:
The calculator also reports the natural residual:
It solves affine variational inequalities over box-constrained sets. That means the operator is written as F(x) = Ax + b, and every variable stays between chosen lower and upper bounds.
Projection forces every updated variable back into its feasible interval. If a component moves below its lower bound or above its upper bound, the calculator clips it to the nearest allowed value.
Start with a modest positive value such as 0.1 or 0.25. If iterations oscillate or stall, reduce the step size. If progress is very slow, try a slightly larger value carefully.
The scaled residual is ||x(k+1) − x(k)|| divided by the step size. Smaller values indicate the iteration is stabilising near a point that satisfies the projected optimality condition.
The natural residual measures how close the final point is to a fixed point of the projection map x = P(K)(x − F(x)). It gives another useful convergence check.
The calculator validates dimensions before solving. Matrix A must be square, and vectors b, bounds, and initial x must all have the same length as the matrix size.
No. Convergence depends on operator properties, step size choice, and the feasible set. Monotone or well-scaled problems usually behave better than poorly conditioned or unstable formulations.
Yes. Enter any square matrix size you need, as long as the vectors and bounds match. Larger systems may require more iterations and careful tuning of the step size.
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.