Numerical experimentation · two objectives, three algorithms
Feasible does not mean nearest
Start at (2, −1), with A = {y ≥ 0} and B = {y ≥ x}. Alternating projections (orange) stop at (1, 1). Dykstra (teal) reaches the nearest point (½, ½); Douglas–Rachford’s shadow (purple) need not.
Dotted black line: the direct nearest-point projection. Symbols mark complete cycles; lines include the intermediate projections for POCS and Dykstra.
A Sudoku solved by reflections
Five copies of 729 real scores evolve in product space. The board below is only a rounded diagnostic, not the solver state. No search or backtracking is used.
Paused · iteration 0
Dark teal: fixed clue. Red and underlined: repeated digit in a row, column or block. Fainter digits: a small top-two score gap. Every digit has an accessible row/column label.
- Iteration
- 0
- Unit violations V
- —
- Shadow disagreement r
- —
- Unrounded clue residual
- —
- Exact solution check
- Not solved
Rounded unit violations V
Unrounded residuals (log₁₀(1 + value))
Consensus disagreement · Clue mismatch
Paused by default; playback stops when the page is hidden, at an exactly checked solution, or at 5,000 iterations. Restarting a seed reproduces the same initialization. There is no convergence guarantee for these nonconvex constraint sets. Finding a valid rounded board does not imply that the real-valued iterates have converged.