FA-12016 / Optimization solver contracts / Open access

Conjugate gradient search direction uses the residual delta · case 01

Conjugate gradient search direction uses the residual delta.

Verified by executionVariant 1 · 6 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Direction recurrence substitutes new-old residual for the new residual.

VERIFIED REPAIR

Use p_new=r_new+beta*p_old for the residual b-Ax convention.

Unsuccessful approach: Using a negative new residual applies the gradient sign convention to a residual recurrence.

Case contract

Return one CG direction vector given residual b-Ax, old direction and nonnegative beta; lengths agree.

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

N = 1
observations = []
def solve(residual, old_residual, direction, beta):
    return [r-o+beta*p for r,o,p in zip(residual,old_residual,direction)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('nonzero history', solve([N], [2*N], [4*N], 0.5), [3*N])
check('restart', solve([N,-N], [1,1], [9,9], 0), [N,-N])
check('zero residual', solve([0], [N], [2*N], 0.5), [N])
check('orthogonal coordinates', solve([N,0], [0,N], [0,2*N], 0.5), [N,N])
check('negative direction', solve([N], [N], [-2*N], 1), [-N])
check('empty vectors', solve([], [], [], 1), [])
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
nonzero history	`[1.0]`	`[3]`	Failed
restart	`[0, -2]`	`[1, -1]`	Failed
zero residual	`[0.0]`	`[1]`	Failed
orthogonal coordinates	`[1.0, 0.0]`	`[1, 1]`	Failed
negative direction	`[-2]`	`[-1]`	Failed
empty vectors	`[]`	`[]`	Passed

SHA-256 / b626a817d7b8a0ef2942ea8626631e1bae578e458c402ecebb29e59fa481050d

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(residual, old_residual, direction, beta):
    return [-r+beta*p for r,p in zip(residual,direction)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('nonzero history', solve([N], [2*N], [4*N], 0.5), [3*N])
check('restart', solve([N,-N], [1,1], [9,9], 0), [N,-N])
check('zero residual', solve([0], [N], [2*N], 0.5), [N])
check('orthogonal coordinates', solve([N,0], [0,N], [0,2*N], 0.5), [N,N])
check('negative direction', solve([N], [N], [-2*N], 1), [-N])
check('empty vectors', solve([], [], [], 1), [])
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
nonzero history	`[1.0]`	`[3]`	Failed
restart	`[-1, 1]`	`[1, -1]`	Failed
zero residual	`[1.0]`	`[1]`	Passed
orthogonal coordinates	`[-1.0, 1.0]`	`[1, 1]`	Failed
negative direction	`[-3]`	`[-1]`	Failed
empty vectors	`[]`	`[]`	Passed

SHA-256 / 76918154f15487de065c4d7881be7a6e18ddbb25fbd6c7c7ae71174081333dbc

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(residual, old_residual, direction, beta):
    return [r+beta*p for r,p in zip(residual,direction)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('nonzero history', solve([N], [2*N], [4*N], 0.5), [3*N])
check('restart', solve([N,-N], [1,1], [9,9], 0), [N,-N])
check('zero residual', solve([0], [N], [2*N], 0.5), [N])
check('orthogonal coordinates', solve([N,0], [0,N], [0,2*N], 0.5), [N,N])
check('negative direction', solve([N], [N], [-2*N], 1), [-N])
check('empty vectors', solve([], [], [], 1), [])
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
nonzero history	`[3.0]`	`[3]`	Passed
restart	`[1, -1]`	`[1, -1]`	Passed
zero residual	`[1.0]`	`[1]`	Passed
orthogonal coordinates	`[1.0, 1.0]`	`[1, 1]`	Passed
negative direction	`[-1]`	`[-1]`	Passed
empty vectors	`[]`	`[]`	Passed

SHA-256 / 849cce9422fdde72c04947148445cfb414c9d32b2e3d19acd5826c452fdd7218

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 2026-09-29T14:38:53.038340+00:00.

Case digest / d848915f5b468704ddcd4239f51f3d8f953b70f87c7e8f6f28e662111f8a338e