Trust: audits, certificates, polishing

Ledge's policy is that you should never have to trust the solver's own claim of success. Every mechanism below evaluates on the original, unscaled problem data and is independently checkable.

Independent KKT audits

check_kkt recomputes primal and dual residuals from the returned point and multipliers — including the L1 subgradient conditions — without using any solver state. The reported primal_residual / dual_residual on every solution come from this audit, never from internal scaled-space estimates. The comparison harness applies the same audit to OSQP and Clarabel results, so published cross-solver numbers use one referee.

Automatic scaling (default on)

Ten Ruiz equilibration passes balance the problem before iterating, preserving factor structure (the scaling acts on rows of F and on d; no dense matrix is formed). Scaling changes only the space ADMM iterates in: termination and all reported residuals are evaluated on original data. scaling_iterations=0 disables it.

Solution polishing (default on)

After convergence, the active set is guessed from the final iterate and one direct KKT solve refines the solution — typically from ~1e-5 residuals to 1e-11 or better, for single-digit-percent extra time. The polished candidate is adopted only if the audited worst KKT residual improves; degenerate active sets are rejected and the ADMM iterate returned unchanged, so polish never degrades a solution and never ships uncertified multipliers. SolveResult.polished / Solution::polished records the outcome.

Infeasibility certificates

Contradictory constraints do not burn 10 000 iterations. The solver detects divergence directions and returns PrimalInfeasible (with a normalized Farkas certificate) or DualInfeasible (with an unbounded descent ray). Certificates are attached to the solution, auditable with check_primal_certificate / check_dual_certificate, and translated into portfolio vocabulary — e.g. "budget row conflicts with the sector caps" — as the leading hint.

The default infeasibility_tolerance=1e-5 is deliberately stricter than OSQP's, because a false "your portfolio is infeasible" is worse than a slow MaxIterations: problems infeasible by a smaller margin fall back to MaxIterations with hints.

In Python, infeasible statuses raise by default with the hint as the message; pass raise_on_failure=False to inspect SolveResult.certificate.

Unconverged solves

MaxIterations results carry convergence_hints: which residual dominates, whether the penalty was re-tuned repeatedly, and what to try (more iterations, scaling, looser tolerance). The final iterate is still returned with honestly reported residuals.