Sudoku as an
optimization laboratory

One Sudoku goal, several learning signals

Sudoku is not inherently a multi-objective decision problem: every legal completion is optimal. Multi-objectivization deliberately transforms one correctness penalty into a vector of related objectives so population search can distinguish candidates that a single total score would tie. The goal is better credit assignment and diversity during search—not a permanent trade-off between imperfect final answers.

For example, (2,8) and (5,5) have the same total penalty but encode different search directions. Pareto dominance keeps both only when neither is better on every objective; crowding preserves representatives from different regions, and the external archive records non-dominated grids across generations. Once any objective vector reaches zero in every component, the certificate is exact and search may stop.

From two objectives to many objectives

• 2D · Row × Column — the clearest baseline and easiest Pareto front to interpret
• 6D · Bands × Stacks — three horizontal and three vertical regions localize conflicts
• 9D · Block responsibility — credits each block for the row/column conflicts caused by its cells
• 9D · Digit conflicts — reveals whether a particular digit remains difficult to place
• 2D · Worst × Total — balances repairing the worst group against overall progress

More dimensions are hypotheses, not automatic improvements. As dimensionality grows, many candidates become non-dominated and selection pressure can weaken. Compare methods with identical populations, seeds and evaluation budgets, then report solution rate, evaluations-to-solution, convergence, spread, archive size and runtime.