FA-12001 / Optimization solver contracts / Open access

Quasi Newton update admits nonpositive curvature · case 01

Quasi Newton update admits nonpositive curvature.

Verified by executionVariant 1 · 9 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The secant pair is used whenever its dot product is nonzero.

VERIFIED REPAIR

Compute the dot product exactly from represented inputs and accept only a strictly positive result, avoiding overflow and cancellation in the sign test.

Unsuccessful approach: Taking an absolute value before testing positivity admits negative curvature.

Case contract

Return whether a finite Python integer/float vector pair may enter an undamped BFGS update under the strict curvature condition s dot y>0, evaluated exactly for the represented inputs; equal vector lengths are assumed.

Why this case matters

This deterministic solver-step model isolates an algorithmic invariant used by iterative optimization implementations.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
N = 1
observations = []
def solve(s, y):
    return sum(a*b for a,b in zip(s,y)) != 0
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('negative pair', solve([N], [-1]), False)
check('positive pair', solve([N], [1]), True)
check('zero step', solve([0], [N]), False)
check('orthogonal', solve([1,0], [0,N]), False)
check('cancellation', solve([1,1], [N,-N]), False)
check('mixed positive', solve([1,2], [-N,N]), True)
check('mixed negative', solve([2,1], [-N,N]), False)
check('overflow cancellation retains positive curvature', solve([1e308,1e308,N],[2.0,-2.0,1.0]), True)
check('overflow cancellation retains negative curvature', solve([1e308,1e308,N],[2.0,-2.0,-1.0]), False)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
negative pair	`True`	`False`	Failed
positive pair	`True`	`True`	Passed
zero step	`False`	`False`	Passed
orthogonal	`False`	`False`	Passed
cancellation	`False`	`False`	Passed
mixed positive	`True`	`True`	Passed
mixed negative	`True`	`False`	Failed
overflow cancellation retains positive curvature	`True`	`True`	Passed
overflow cancellation retains negative curvature	`True`	`False`	Failed

SHA-256 / 68b9ef59f95af0d13fb703a46249f6f9eaf3c1b1a653edb4141e924ce8cda49c

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
N = 1
observations = []
def solve(s, y):
    return abs(sum(a*b for a,b in zip(s,y))) > 0
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('negative pair', solve([N], [-1]), False)
check('positive pair', solve([N], [1]), True)
check('zero step', solve([0], [N]), False)
check('orthogonal', solve([1,0], [0,N]), False)
check('cancellation', solve([1,1], [N,-N]), False)
check('mixed positive', solve([1,2], [-N,N]), True)
check('mixed negative', solve([2,1], [-N,N]), False)
check('overflow cancellation retains positive curvature', solve([1e308,1e308,N],[2.0,-2.0,1.0]), True)
check('overflow cancellation retains negative curvature', solve([1e308,1e308,N],[2.0,-2.0,-1.0]), False)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
negative pair	`True`	`False`	Failed
positive pair	`True`	`True`	Passed
zero step	`False`	`False`	Passed
orthogonal	`False`	`False`	Passed
cancellation	`False`	`False`	Passed
mixed positive	`True`	`True`	Passed
mixed negative	`True`	`False`	Failed
overflow cancellation retains positive curvature	`False`	`True`	Failed
overflow cancellation retains negative curvature	`False`	`False`	Passed

SHA-256 / f860649c73083b7a2c1a1a7211da5c12b6abd8db33e054043359c6b2d3bb16f6

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
N = 1
observations = []
def solve(s, y):
    return sum(Fraction(a)*Fraction(b) for a,b in zip(s,y)) > 0
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('negative pair', solve([N], [-1]), False)
check('positive pair', solve([N], [1]), True)
check('zero step', solve([0], [N]), False)
check('orthogonal', solve([1,0], [0,N]), False)
check('cancellation', solve([1,1], [N,-N]), False)
check('mixed positive', solve([1,2], [-N,N]), True)
check('mixed negative', solve([2,1], [-N,N]), False)
check('overflow cancellation retains positive curvature', solve([1e308,1e308,N],[2.0,-2.0,1.0]), True)
check('overflow cancellation retains negative curvature', solve([1e308,1e308,N],[2.0,-2.0,-1.0]), False)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
negative pair	`False`	`False`	Passed
positive pair	`True`	`True`	Passed
zero step	`False`	`False`	Passed
orthogonal	`False`	`False`	Passed
cancellation	`False`	`False`	Passed
mixed positive	`True`	`True`	Passed
mixed negative	`False`	`False`	Passed
overflow cancellation retains positive curvature	`True`	`True`	Passed
overflow cancellation retains negative curvature	`False`	`False`	Passed

SHA-256 / 37d9b02d02408212e240b998f15a2acf87042cc807497ff7ff41b2796f0cef47

Verification & scope

Controlled finite inputs and explicit one-step contracts; this is not a production solver or a numerical stability benchmark. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:38:52.960065+00:00.

Case digest / 13aac874718a9fa87e3cb61d756dce1ac5435b0f01c60c9ab8d0774ba31c6f36