Continuous LP observations: O1

solve_lp_observed returns an ordinary owning SolveResult plus an immutable shared_ptr<const LpObservations>. SolveSession has the same overloads. The observation artifact owns the exact historical ModelSnapshot; source handles, revision, names and tombstones survive edits, reset, and model/session destruction. No SolveResult layout or ordinary solve API changed.

This first implementation accepts original Continuous linear models through HiGHS with Guarantee::Numerical. Auto selects that same backend. Native, Exact, Certified, any active integer/binary/semi variable (including fixed integers), indicators, and native globals return explicit Unsupported; no relaxation, conversion of variable types, or automatic fallback occurs. The model is validated and admitted before a persistent backend is mutated. Missing HiGHS is also explicit Unsupported. A malformed model or observation tolerance is InvalidModel.

#include <gecode/optimize/lp_observations.hpp>
#include <gecode/optimize/session.hpp>
using namespace Gecode::Optimize;
Model model;
auto x = model.add_continuous();
auto y = model.add_continuous();
auto demand = model.add_row({{x, 1}, {y, 1}}, 4,
                           std::numeric_limits<double>::infinity());
model.minimize({{x, 2}, {y, 3}}, 7);
auto observed = solve_lp_observed(model);
// Check the solve status and each group's state before using optional fields.
if (observed.observations &&
    observed.observations->dual_point().state == LpObservationState::Available) {
  double demand_dual = *observed.observations->row(demand).dual; // 2
  double y_reduced_cost = *observed.observations->column(y).reduced_cost; // 1
  (void)demand_dual; (void)y_reduced_cost;
}

Groups and original coordinates

The primal row, dual point, and basis groups independently report NotRequested, Available, Unavailable, or Rejected with a typed reason and message. Missing data is never represented by zero. Raw vector length/identity errors and failed numerical checks are distinguished from absent backend data. Invalid input or allocation failure can prevent creating an artifact altogether.

• rows() and columns() retain original slot counts, including inactive tombstones. Handle accessors reject foreign IDs, tombstones, and out-of-range slots. Backend row/column maps are checked against the exact active source.
• Original row activities and signed lower/upper slacks are recomputed from the source matrix and validated primal values. An infinite bound side has no slack. Tiny negative slacks remain signed; they are not silently clipped.
• Duals and reduced costs satisfy the original objective convention A^T pi + r = c, for both minimization and maximization. HiGHS already restores its public solution to the passed model after presolve and objective-sense conversion. The adapter scatters those coordinates back to original slots.
• A feasible constant row omitted by backend conversion still receives an independently recomputed activity/slacks and a zero dual marked DerivedConstantRow. The other row duals are marked Backend.
• Basis status is a typed copy, with original-bound position, dimension, and basic-count checks. It requires both backend basis and info validity flags. The entire basis group is unavailable if an active constant row was elided; the implementation does not invent a full original basis. No independent nonsingularity or numerical conditioning claim is made.

Primal row observations require an independently validated finite source point. Duals and basis initially require a timely final Optimal primal point and valid backend info/primal flags; duals additionally require valid feasible dual flags. An infeasible or unbounded solve does not produce an O1 ray or dual certificate. An empty model solved without a backend call can have an available empty primal row group, while its dual and basis groups are unavailable (NoBackendSolve). duals=false and basis=false suppress their respective requests/publication.

Independent numerical KKT checks

The checker uses original rows, costs, bounds and source values, rather than the backend's presolved/packed matrix or its reported residuals. It checks:

1. Original primal feasibility and objective consistency.
2. Dual normal-cone signs at finite row/column bounds.
3. Stationarity s*c - A^T(s*pi) - s*r, where s is +1 for min and -1 for max.
4. Complementarity against the endpoint selected by each signed multiplier.
5. The signed normalized primal-minus-dual gap, with the common objective offset cancelled before accumulation.

Accumulation uses checked compensated long double arithmetic. Uncollapsed sum and correction components flow into slack, stationarity, complementarity and basis checks, including platforms where long double has double precision. A nonzero multiplier requiring an infinite endpoint prevents a finite dual objective/gap; the checker never treats that product as zero. Overflow, nonfinite entries, malformed dimensions or coordinate maps reject publication.

Check tolerances are explicit, finite, nonnegative absolute quantities in original units. Defaults are 1e-7 for dual feasibility and stationarity, 1e-6 for complementarity and objective gap. The primal check uses the solve's feasibility tolerance. Effective backend primal/dual tolerances are recorded when obtained; they do not replace the independently requested check tolerances. Residuals and signed gap are retained without clipping, including a small negative gap accepted within absolute tolerance. Large objective offsets cannot hide a meaningful gap by rounding both reported objective scalars together.

These are numerical observations, not exact certificates. An accepted KKT report and its dual_objective_estimate do not strengthen the ordinary result's guarantee or overwrite its backend best bound. A rejected dual or basis group also does not erase an otherwise valid ordinary primal solve result.

Budgets and repeated sessions

One shared monotonic budget starts before source snapshot copying and covers conversion, backend execution, passive collection, independent checks, and raw capture destruction. Public getters do no solver work. The collector uses only current getSolution, getInfo, getBasis, and option values; it does not call ray, ranging, basis factorization or extra optimization routines.

O1 chooses a conservative whole-call publication policy: if the final budget checkpoint detects cancellation/time/node exhaustion, all observation groups are cleared, with requested groups Unavailable/Interrupted and NotRequested groups unchanged. This also clears groups that completed earlier. The timely independently validated ordinary primal incumbent is retained, and the solve's termination reflects the limit. The final gate runs after raw capture vectors have been destroyed. A private deterministic test cancels at this boundary, without sleeps or production hooks. Budget enforcement is cooperative; memory allocation/destruction and solver calls are not forcibly preempted.

Every call creates a new artifact and new raw vectors. Limited/error solves cannot expose the prior call's duals or basis. Session edits compare source content and compact mappings, not only revision numbers; updated costs, bounds, rows, sense or offset are reflected in the new artifact. Session basis reuse remains internal and does not imply that the new result has an exportable basis.

Verification and remaining scope

test/optimize/lp_observations.cpp checks real HiGHS analytic min/max/ranged-row solutions, constant-row scattering, tombstones and duplicate names, historical ownership, same-revision snapshot edits, repeated observed/ordinary session calls, disabled groups, missing backend/admission, infeasible and unbounded models, cancellation and malformed input.

test/optimize/lp_observations_checks.cpp exercises the pure private checker with known primal/dual data and corrupted flags, IDs, revisions, masks, vector lengths, NaNs, infinities, basis statuses, stationarity/sign/complementarity/gap failures, large-offset and 1e16 + 1 - 1e16 cancellation, and post-collection cancellation. It never substitutes fake data into a production solve. The private detail header is not installed.

The separate basis submission API now provides owning basis starts; the observation operation itself remains passive. Rays, sensitivity/ranging, MIP relaxation observations, QP observations and exact dual proofs remain outside O1. The subsequent research and implementation boundaries are in LP-OBSERVATIONS-DESIGN.md. Bindings use separate owning observed-result handles and copied ordinary results; the public C++ artifact contains no borrowed backend memory.

The C/Python guide documents the installed binding surface. One FAST case checks analytic min/max objectives, original duals, reduced costs, slacks and basis/KKT availability. Completed integrated and installed-consumer checks are recorded in validation.