make_lp_basis copies either an available O1 basis or caller-created LpBasisData into an immutable owning LpBasis. solve_lp_with_basis and its SolveSession overload return an ordinary observed solve plus a separate submission report and an owning reference to the requested basis. Existing SolveOptions, LpObservationOptions, and O1 record layouts are unchanged.

using namespace Gecode::Optimize;
auto first = solve_lp_observed(model);
if (first.observations &&
    first.observations->basis().state == LpObservationState::Available) {
  LpBasisSolveOptions options;
  options.basis = make_lp_basis(*first.observations);
  auto next = solve_lp_with_basis(model, options);
  // next.submission describes submission; next.observed.result describes solve.
}

Initial source and status policy

A basis source must be an ordinary Continuous linear model with no active indicator/global records or constant rows. Each row/column status vector has exactly the original slot count; live slots have a status, tombstones have none. Every enum is checked explicitly. Lower/Upper require the corresponding finite bound; Zero requires a free row/column. Fixed entities may be nonbasic on either side. NonbasicUnspecified lets HiGHS choose its nonbasic position. Basic statuses are permitted on any entity, but the total number of basic entities must equal the number of active rows. These structural checks do not claim nonsingularity, primal feasibility, dual feasibility, or optimality. A valid nonoptimal basis is a legitimate start.

Caller data is copied before the factory returns. Changes to its source or status arrays cannot alter the artifact. Basis origin records whether the factory used caller data or an O1 observation. An unavailable O1 basis cannot be converted into an empty or fabricated start.

The first submission policy requires the same source model owner, revision, original variable/row slot identities and active masks. Active variable types, bounds and names; active row bounds, names and matrix terms; and objective sense, offset and coefficients must match exactly under ordinary scalar/string equality. The same revision alone is insufficient: public snapshots are untrusted. Inactive payloads do not describe the mathematical model and need not match; inactive variable/row identities and masks still must match. Every complete input snapshot is independently structurally validated, including original protected metadata. Active indicators/globals remain unsupported.

Changing a cost, bound, row side, sense, offset or active name/revision requires a new basis artifact. More permissive compatible-matrix hint policies are deferred. The existing internal session basis reuse policy is unchanged and is separate from this public source-matched submission contract. Foreign source owners, malformed dimensions/statuses and source mismatches fail before backend/session mutation. A missing basis is an explicit InvalidModel input error; the API never silently turns into an ordinary solve. Simultaneous primal_start is rejected rather than assigning implicit precedence.

Numerical HiGHS is the only supported backend/guarantee. Native, Exact and Certified remain explicit Unsupported. A missing HiGHS build returns Unsupported without a backend attempt. Basis factories perform structural checks without requiring an installed numerical backend. Empty models can carry a valid empty basis; the existing direct empty-model solve performs no basis submission and reports NotAttempted.

What submission states mean

NotAttempted: admission, unsupported routing, a pre-submission limit, or an empty model prevented a backend basis call. The ordinary result explains why.
Accepted: HiGHS accepted and returned structurally valid statuses identical to the request, before optimization.
Repaired: HiGHS accepted but changed at least one status during its immediate basis factorization/repair step. This includes completing/replacing singular structural columns with logical row columns, and any nonbasic normalization.
Rejected: backend submission or validation of its returned status data failed. Allocation failure also prevents completing submission and has the separate ordinary MemoryLimit termination. No fallback optimization occurs.
Interrupted: the submission/checking phase crossed its cooperative budget; complete accepted-status information is not published.

backend_attempted records whether setBasis was entered. statuses_changed is absent until the accepted returned statuses have been completely checked; false is distinct from absent. An accepted/repaired submission can be followed by a limited, infeasible, unbounded or failed solve. It is not evidence that the final solve is optimal or faster. Changes to the final optimum basis do not retroactively turn Accepted into Repaired. Earlier submission facts may survive a later solve limit, while O1 follows its own conservative final-limit clearing policy.

The result retains the requested basis and its original source independently of the input options, model and session. Final observations and ordinary result are separate owning historical artifacts. No checkpoint, saved search tree, exact proof, conditioning guarantee or independent nonsingularity certificate is provided.

Backend mechanics, cleanup and budget

Pinned HiGHS 1.15.1 checks and factors nonzero-row alien bases inside setBasis, which may repair singular or incomplete bases. O2 always uses that alien path for nonzero rows, even when the source artifact came from HiGHS observations. It does not trust a caller-provided validity flag. Returned statuses are inspected and compared immediately after setBasis, before run. All compact slots are mapped explicitly to original slots and revalidated. HiGHS setBasis, alien basis accommodation.

The pinned alien zero-row branch indexes internal column-status storage before its general dimension check. O2 avoids that branch: zero rows require zero basic entities and exact dimensions, so the basis matrix has dimension zero; a checked non-alien basis uses HiGHS's consistency path. This limited exception does not trust a potentially singular nonzero-dimensional caller basis.

The shared monotonic budget starts before source copying and covers admission, conversion, basis preparation, factorization/repair, optimization, O1 checks and cleanup. Interrupt callbacks and the remaining HiGHS time limit are installed before submission. Remaining time is reapplied before optimization without restarting the outer clock. Factorization/scaling may contain uninterruptible regions; checks around those regions prevent a late phase from publishing a false completion but do not guarantee forced wall-clock preemption.

Rejected or interrupted submission unwinds the callback scope before discarding the dirty backend. The next session call loads a clean model. A backend rejection does not trigger a hidden cold solve. Ordinary subsequent solves reset their per-call options and work normally. Explicit submitted bases are not counted as an internally retained basis_warm_starts statistic. Other model-load/update statistics retain their existing meanings.

Verification and build integration

test/optimize/lp_basis.cpp covers copied factory ownership, malformed IDs/statuses, missing/extra slots, invalid finite-side/zero statuses, incorrect basic counts, all active-content/revision compatibility gates, missing starts/backend, unsupported policies, simultaneous primal starts, a nonoptimal accepted basis whose final optimum basis differs, singular repair, safe zero-row submission, tombstones, ordinary/observed session reuse, disabled O1 groups and pre-cancellation.

test/optimize/lp_basis_failure.cpp is a separately compiled test variant. Compile solve.cpp with GECODE_OPTIMIZE_WITH_HIGHS=1 and GECODE_OPTIMIZE_TEST_LP_BASIS_FAILURE=1, and link this test with the remaining ordinary Optimize objects plus real HiGHS. The test supplies the private selector Detail::lp_basis_test_cancel(). Immediately after a successful real setBasis mutation, the variant injects either a backend error result or cancellation. It checks no fallback, no stale observation publication, callback/dirty-state cleanup, and a successful next ordinary solve with a new model load. The variant also covers zero-row rejection. Its selector and injection code are absent from production binaries; the normal test uses unmodified backend return values.

Normal and fully instrumented facade/HiGHS ASan+UBSan panels are required, along with backend-free admission tests and O1/ordinary solve/session regression tests. lp_basis_detail.hpp is private and must not be installed. Root build registration and later C/Python bindings remain separate integration work. Rays, basis solves, sensitivity/ranging and compatible-edited-source hints remain outside O2.