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## Problem Maximizing/minimizing under a **strict** inequality has a delta-rational optimum. For ```smt2 (declare-const r Real) (assert (< r 1)) (maximize r) (check-sat) (get-objectives) ``` the optimum is the supremum `1 - epsilon`, but z3 reported `r = 0`. The same defect makes shared-symbol objectives report a value matching **neither the model nor the true optimum** (issue #10028 follow-up). Minimal reproducer — a 6-mark Golomb ruler (a `>32`-arg `distinct`, so the objective is coupled to EUF) with a strict real objective `obj > x5`, whose true optimum is `17 + epsilon`: | case | before | after | |---|---|---| | `maximize r`, `r < 1` | `0` ❌ | `1 - epsilon` ✅ | | `minimize r`, `r > 1` | `0` ❌ | `1 + epsilon` ✅ | | Golomb `minimize obj`, `obj > x5` | `35/2` / `7+eps` ❌ | `17 + epsilon` ✅ | ## Root cause `check_bound` validates the LP hint by asserting `objective >= optimum`. For a supremum `1 - epsilon` this is a **lower** bound whose value carries a **negative** infinitesimal `(1, -1)`. No `lconstraint_kind` can express that. The kind->infinitesimal map only yields the *matching-sign* cases — `GT` -> lower `(r, +1)`, `LT` -> upper `(r, -1)` — or zero (`GE`/`LE`). The opposite-sign lower bound `(r, -1)` (i.e. `r >= r0 - delta`) is a *relaxation* that no strict inequality produces. `opt_solver::mk_ge` therefore projected the `-epsilon` away, turning `r >= 1 - epsilon` into the over-strong, unsatisfiable `r >= 1`; validation failed and the strictly smaller current model value was reported instead. ## Fix — carry the infinitesimal faithfully through the bound pipeline - **`lp_api::bound`** gains an `eps` component so `get_value` returns the true delta value (no spurious rational fixed-variable equality is propagated to EUF). - **`lar_base_constraint`** stores its right-hand side as a delta-rational `impq` pair; `rhs()` returns the rational component, `bound_eps()` the infinitesimal one. - **`lar_solver`** bound activation/update threads the whole `impq` bound, so a lower bound `(r, -1)` can be asserted. `constraint_holds` accounts for it using the **same** strict-bounds delta that flattens the model, computed **once per model**. - **`theory_lra::mk_ge`** builds a *fresh* predicate for the `(r, -1)` lower bound (to avoid colliding with an already-internalized `v >= r` literal) and attaches `eps = -1`. **`opt_solver::mk_ge`** passes the unprojected value to `theory_lra` / `theory_mi_arith` / `theory_inf_arith` (whose bounds are already `inf_rational`). The pair machinery is what makes the supremum both representable (optimum `1 - epsilon`) and validatable; the reported witness model remains the flattened rational (`find_delta_for_strict_bounds`), consistent with the existing epsilon semantics. ## Validation - Strict optima correct: `1-eps`, `1+eps`, bounded `2<r<5 -> 5-eps`, and lex/box variants. - Integer optima and the #10028 shared-symbol cases unchanged (Golomb n=6/7/8 -> 17/25/34, consistent with the model). - Unit tests **92/92** (release); no new debug-suite failures. - Opt regression corpus (73 files, `model_validate=true`) **byte-identical** to baseline. Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com> --------- Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com> Co-authored-by: Nikolaj Bjorner <nbjorner@microsoft.com> |
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