# 梯度下降法基礎 · 脈衝 R03

本批已通過 Schema、去重、來源快照與獨立答案重算；仍未標記為人工審定。

本題庫已有對應的中英逐步詳解。原題的 candidate 與英文 ai_candidate 狀態保留；AI 覆核、有限計算與 Lean 形式證明分開標示。

batch-20261015-advance-20261119-r03-gradient-descent-foundations · 2026-10-15 · 100 題

欄序為來源綁定的呈現補充；原始 JSONL 與數學答案不變。

[欄序與來源綁定資料](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[1,2],"linear":[-1,-2],"point":[-2,-3]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([1, 2], [-1, -2], [-2, -3], [-3, -8], 19, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([1, 2], [-1, -2], [-2, -3], [-3, -8], 19, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0001/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 2. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[2,3],"linear":[0,-1],"point":[-1,-2]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([2, 3], [0, -1], [-1, -2], [-2, -7], 9, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([2, 3], [0, -1], [-1, -2], [-2, -7], 9, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0002/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 3. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[3,4],"linear":[1,0],"point":[0,-1]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([3, 4], [1, 0], [0, -1], [1, -4], 2, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([3, 4], [1, 0], [0, -1], [1, -4], 2, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0003/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 4. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[1,5],"linear":[2,1],"point":[1,0]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([1, 5], [2, 1], [1, 0], [3, 1], 5/2, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([1, 5], [2, 1], [1, 0], [3, 1], 5/2, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0004/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 5. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[2,2],"linear":[-2,2],"point":[2,1]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([2, 2], [-2, 2], [2, 1], [2, 4], 3, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([2, 2], [-2, 2], [2, 1], [2, 4], 3, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0005/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 6. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[3,3],"linear":[-1,3],"point":[-2,2]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([3, 3], [-1, 3], [-2, 2], [-7, 9], 20, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([3, 3], [-1, 3], [-2, 2], [-7, 9], 20, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0006/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 7. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[1,4],"linear":[0,-3],"point":[-1,3]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([1, 4], [0, -3], [-1, 3], [-1, 9], 19/2, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([1, 4], [0, -3], [-1, 3], [-1, 9], 19/2, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0007/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 8. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[2,5],"linear":[1,-2],"point":[0,-3]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([2, 5], [1, -2], [0, -3], [1, -17], 57/2, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([2, 5], [1, -2], [0, -3], [1, -17], 57/2, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0008/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 9. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[3,2],"linear":[2,-1],"point":[1,-2]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([3, 2], [2, -1], [1, -2], [5, -5], 19/2, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([3, 2], [2, -1], [1, -2], [5, -5], 19/2, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0009/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 10. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[1,3],"linear":[-2,0],"point":[2,-1]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([1, 3], [-2, 0], [2, -1], [0, -3], -1/2, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([1, 3], [-2, 0], [2, -1], [0, -3], -1/2, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0010/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 11. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[2,4],"linear":[-1,1],"point":[-2,0]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([2, 4], [-1, 1], [-2, 0], [-5, 1], 6, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([2, 4], [-1, 1], [-2, 0], [-5, 1], 6, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0011/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 12. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[3,5],"linear":[0,2],"point":[-1,1]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([3, 5], [0, 2], [-1, 1], [-3, 7], 6, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([3, 5], [0, 2], [-1, 1], [-3, 7], 6, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0012/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 13. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[1,2],"linear":[1,3],"point":[0,2]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([1, 2], [1, 3], [0, 2], [1, 7], 10, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([1, 2], [1, 3], [0, 2], [1, 7], 10, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0013/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 14. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[2,3],"linear":[2,-3],"point":[1,3]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([2, 3], [2, -3], [1, 3], [4, 6], 15/2, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([2, 3], [2, -3], [1, 3], [4, 6], 15/2, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0014/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 15. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[3,4],"linear":[-2,-2],"point":[2,-3]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([3, 4], [-2, -2], [2, -3], [4, -14], 26, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([3, 4], [-2, -2], [2, -3], [4, -14], 26, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0015/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 16. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[1,5],"linear":[-1,-1],"point":[-2,-2]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([1, 5], [-1, -1], [-2, -2], [-3, -11], 16, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([1, 5], [-1, -1], [-2, -2], [-3, -11], 16, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0016/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 17. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[2,2],"linear":[0,0],"point":[-1,-1]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([2, 2], [0, 0], [-1, -1], [-2, -2], 2, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([2, 2], [0, 0], [-1, -1], [-2, -2], 2, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0017/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 18. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[3,3],"linear":[1,1],"point":[0,0]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([3, 3], [1, 1], [0, 0], [1, 1], 0, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([3, 3], [1, 1], [0, 0], [1, 1], 0, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0018/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 19. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[1,4],"linear":[2,2],"point":[1,1]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([1, 4], [2, 2], [1, 1], [3, 6], 13/2, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([1, 4], [2, 2], [1, 1], [3, 6], 13/2, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0019/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 20. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[2,5],"linear":[-2,3],"point":[2,2]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([2, 5], [-2, 3], [2, 2], [2, 13], 16, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([2, 5], [-2, 3], [2, 2], [2, 13], 16, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0020/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 21. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[3,2],"linear":[-1,-3],"point":[-2,3]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([3, 2], [-1, -3], [-2, 3], [-7, 3], 8, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([3, 2], [-1, -3], [-2, 3], [-7, 3], 8, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0021/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 22. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[1,3],"linear":[0,-2],"point":[-1,-3]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([1, 3], [0, -2], [-1, -3], [-1, -11], 20, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([1, 3], [0, -2], [-1, -3], [-1, -11], 20, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0022/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 23. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[2,4],"linear":[1,-1],"point":[0,-2]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([2, 4], [1, -1], [0, -2], [1, -9], 10, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([2, 4], [1, -1], [0, -2], [1, -9], 10, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0023/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 24. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[3,5],"linear":[2,0],"point":[1,-1]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([3, 5], [2, 0], [1, -1], [5, -5], 6, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([3, 5], [2, 0], [1, -1], [5, -5], 6, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0024/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 25. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={"diagonal":[1,2],"linear":[-2,1],"point":[2,0]}；請依 [positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention] 回答。

查看答案與解釋([1, 2], [-2, 1], [2, 0], [0, 1], -2, fixed_standard_Euclidean_coordinate_order)

Wave6第十三層核對=([1, 2], [-2, 1], [2, 0], [0, 1], -2, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0025/)

梯度下降法 · qf-mod50-gd-quadratic-gradient · candidate





## 26. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[1,2],"linear":[-1,-2],"point":[-2,-3],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([-2, -3], [-3, -8], 1/4, [-5/4, -1], 19, 161/32, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([-2, -3], [-3, -8], 1/4, [-5/4, -1], 19, 161/32, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0026/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 27. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[2,3],"linear":[0,-1],"point":[-1,-2],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([-1, -2], [-2, -7], 1/4, [-1/2, -1/4], 9, 19/32, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([-1, -2], [-2, -7], 1/4, [-1/2, -1/4], 9, 19/32, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0027/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 28. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[3,4],"linear":[1,0],"point":[0,-1],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([0, -1], [1, -4], 1/4, [-1/4, 0], 2, -5/32, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([0, -1], [1, -4], 1/4, [-1/4, 0], 2, -5/32, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0028/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 29. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[1,5],"linear":[2,1],"point":[1,0],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([1, 0], [3, 1], 1/4, [1/4, -1/4], 5/2, 7/16, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([1, 0], [3, 1], 1/4, [1/4, -1/4], 5/2, 7/16, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0029/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 30. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[2,2],"linear":[-2,2],"point":[2,1],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([2, 1], [2, 4], 1/4, [3/2, 0], 3, -3/4, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([2, 1], [2, 4], 1/4, [3/2, 0], 3, -3/4, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0030/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 31. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[3,3],"linear":[-1,3],"point":[-2,2],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([-2, 2], [-7, 9], 1/4, [-1/4, -1/4], 20, -5/16, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([-2, 2], [-7, 9], 1/4, [-1/4, -1/4], 20, -5/16, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0031/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 32. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[1,4],"linear":[0,-3],"point":[-1,3],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([-1, 3], [-1, 9], 1/4, [-3/4, 3/4], 19/2, -27/32, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([-1, 3], [-1, 9], 1/4, [-3/4, 3/4], 19/2, -27/32, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0032/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 33. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[2,5],"linear":[1,-2],"point":[0,-3],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([0, -3], [1, -17], 1/4, [-1/4, 5/4], 57/2, 39/32, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([0, -3], [1, -17], 1/4, [-1/4, 5/4], 57/2, 39/32, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0033/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 34. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[3,2],"linear":[2,-1],"point":[1,-2],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([1, -2], [5, -5], 1/4, [-1/4, -3/4], 19/2, 29/32, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([1, -2], [5, -5], 1/4, [-1/4, -3/4], 19/2, 29/32, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0034/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 35. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[1,3],"linear":[-2,0],"point":[2,-1],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([2, -1], [0, -3], 1/4, [2, -1/4], -1/2, -61/32, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([2, -1], [0, -3], 1/4, [2, -1/4], -1/2, -61/32, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0035/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 36. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[2,4],"linear":[-1,1],"point":[-2,0],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([-2, 0], [-5, 1], 1/4, [-3/4, -1/4], 6, 19/16, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([-2, 0], [-5, 1], 1/4, [-3/4, -1/4], 6, 19/16, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0036/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 37. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[3,5],"linear":[0,2],"point":[-1,1],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([-1, 1], [-3, 7], 1/4, [-1/4, -3/4], 6, 0, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([-1, 1], [-3, 7], 1/4, [-1/4, -3/4], 6, 0, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0037/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 38. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[1,2],"linear":[1,3],"point":[0,2],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([0, 2], [1, 7], 1/4, [-1/4, 1/4], 10, 19/32, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([0, 2], [1, 7], 1/4, [-1/4, 1/4], 10, 19/32, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0038/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 39. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[2,3],"linear":[2,-3],"point":[1,3],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([1, 3], [4, 6], 1/4, [0, 3/2], 15/2, -9/8, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([1, 3], [4, 6], 1/4, [0, 3/2], 15/2, -9/8, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0039/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 40. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[3,4],"linear":[-2,-2],"point":[2,-3],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([2, -3], [4, -14], 1/4, [1, 1/2], 26, -1, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([2, -3], [4, -14], 1/4, [1, 1/2], 26, -1, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0040/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 41. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[1,5],"linear":[-1,-1],"point":[-2,-2],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([-2, -2], [-3, -11], 1/4, [-5/4, 3/4], 16, 43/16, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([-2, -2], [-3, -11], 1/4, [-5/4, 3/4], 16, 43/16, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0041/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 42. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[2,2],"linear":[0,0],"point":[-1,-1],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([-1, -1], [-2, -2], 1/4, [-1/2, -1/2], 2, 1/2, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([-1, -1], [-2, -2], 1/4, [-1/2, -1/2], 2, 1/2, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0042/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 43. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[3,3],"linear":[1,1],"point":[0,0],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([0, 0], [1, 1], 1/4, [-1/4, -1/4], 0, -5/16, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([0, 0], [1, 1], 1/4, [-1/4, -1/4], 0, -5/16, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0043/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 44. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[1,4],"linear":[2,2],"point":[1,1],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([1, 1], [3, 6], 1/4, [1/4, -1/2], 13/2, 1/32, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([1, 1], [3, 6], 1/4, [1/4, -1/2], 13/2, 1/32, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0044/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 45. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[2,5],"linear":[-2,3],"point":[2,2],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([2, 2], [2, 13], 1/4, [3/2, -5/4], 16, -19/32, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([2, 2], [2, 13], 1/4, [3/2, -5/4], 16, -19/32, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0045/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 46. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[3,2],"linear":[-1,-3],"point":[-2,3],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([-2, 3], [-7, 3], 1/4, [-1/4, 9/4], 8, -43/32, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([-2, 3], [-7, 3], 1/4, [-1/4, 9/4], 8, -43/32, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0046/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 47. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[1,3],"linear":[0,-2],"point":[-1,-3],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([-1, -3], [-1, -11], 1/4, [-3/4, -1/4], 20, 7/8, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([-1, -3], [-1, -11], 1/4, [-3/4, -1/4], 20, 7/8, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0047/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 48. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[2,4],"linear":[1,-1],"point":[0,-2],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([0, -2], [1, -9], 1/4, [-1/4, 1/4], 10, -5/16, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([0, -2], [1, -9], 1/4, [-1/4, 1/4], 10, -5/16, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0048/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 49. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[3,5],"linear":[2,0],"point":[1,-1],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([1, -1], [5, -5], 1/4, [-1/4, 1/4], 6, -1/4, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([1, -1], [5, -5], 1/4, [-1/4, 1/4], 6, -1/4, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0049/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 50. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[1,2],"linear":[-2,1],"point":[2,0],"step_size":[1,4]}；請依 [point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope] 回答。

one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。

查看答案與解釋([2, 0], [0, 1], 1/4, [2, -1/4], -2, -35/16, 是, positive_step_alone_does_not_guarantee_descent)

Wave6第十三層核對=([2, 0], [0, 1], 1/4, [2, -1/4], -2, -35/16, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0050/)

梯度下降法 · qf-mod50-gd-one-step · candidate





## 51. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[1,2],"linear":[-1,-2],"point":[-2,-3],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(2, 1/4, 1, 1, [[0, [-2, -3], 19, [-3, -8]], [1, [-5/4, -1], 161/32, [-9/4, -4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(2, 1/4, 1, 1, [[0, [-2, -3], 19, [-3, -8]], [1, [-5/4, -1], 161/32, [-9/4, -4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0051/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 52. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[2,3],"linear":[0,-1],"point":[-1,-2],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(3, 1/4, 2/3, 1, [[0, [-1, -2], 9, [-2, -7]], [1, [-1/2, -1/4], 19/32, [-1, -7/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(3, 1/4, 2/3, 1, [[0, [-1, -2], 9, [-2, -7]], [1, [-1/2, -1/4], 19/32, [-1, -7/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0052/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 53. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[3,4],"linear":[1,0],"point":[0,-1],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(4, 1/4, 1/2, 1, [[0, [0, -1], 2, [1, -4]], [1, [-1/4, 0], -5/32, [1/4, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(4, 1/4, 1/2, 1, [[0, [0, -1], 2, [1, -4]], [1, [-1/4, 0], -5/32, [1/4, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0053/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 54. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[1,5],"linear":[2,1],"point":[1,0],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(5, 1/4, 2/5, 1, [[0, [1, 0], 5/2, [3, 1]], [1, [1/4, -1/4], 7/16, [9/4, -1/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(5, 1/4, 2/5, 1, [[0, [1, 0], 5/2, [3, 1]], [1, [1/4, -1/4], 7/16, [9/4, -1/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0054/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 55. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[2,2],"linear":[-2,2],"point":[2,1],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(2, 1/4, 1, 1, [[0, [2, 1], 3, [2, 4]], [1, [3/2, 0], -3/4, [1, 2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(2, 1/4, 1, 1, [[0, [2, 1], 3, [2, 4]], [1, [3/2, 0], -3/4, [1, 2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0055/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 56. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[3,3],"linear":[-1,3],"point":[-2,2],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(3, 1/4, 2/3, 1, [[0, [-2, 2], 20, [-7, 9]], [1, [-1/4, -1/4], -5/16, [-7/4, 9/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(3, 1/4, 2/3, 1, [[0, [-2, 2], 20, [-7, 9]], [1, [-1/4, -1/4], -5/16, [-7/4, 9/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0056/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 57. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[1,4],"linear":[0,-3],"point":[-1,3],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(4, 1/4, 1/2, 1, [[0, [-1, 3], 19/2, [-1, 9]], [1, [-3/4, 3/4], -27/32, [-3/4, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(4, 1/4, 1/2, 1, [[0, [-1, 3], 19/2, [-1, 9]], [1, [-3/4, 3/4], -27/32, [-3/4, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0057/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 58. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[2,5],"linear":[1,-2],"point":[0,-3],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(5, 1/4, 2/5, 1, [[0, [0, -3], 57/2, [1, -17]], [1, [-1/4, 5/4], 39/32, [1/2, 17/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(5, 1/4, 2/5, 1, [[0, [0, -3], 57/2, [1, -17]], [1, [-1/4, 5/4], 39/32, [1/2, 17/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0058/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 59. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[3,2],"linear":[2,-1],"point":[1,-2],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(3, 1/4, 2/3, 1, [[0, [1, -2], 19/2, [5, -5]], [1, [-1/4, -3/4], 29/32, [5/4, -5/2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(3, 1/4, 2/3, 1, [[0, [1, -2], 19/2, [5, -5]], [1, [-1/4, -3/4], 29/32, [5/4, -5/2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0059/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 60. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[1,3],"linear":[-2,0],"point":[2,-1],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(3, 1/4, 2/3, 1, [[0, [2, -1], -1/2, [0, -3]], [1, [2, -1/4], -61/32, [0, -3/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(3, 1/4, 2/3, 1, [[0, [2, -1], -1/2, [0, -3]], [1, [2, -1/4], -61/32, [0, -3/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0060/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 61. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[2,4],"linear":[-1,1],"point":[-2,0],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(4, 1/4, 1/2, 1, [[0, [-2, 0], 6, [-5, 1]], [1, [-3/4, -1/4], 19/16, [-5/2, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(4, 1/4, 1/2, 1, [[0, [-2, 0], 6, [-5, 1]], [1, [-3/4, -1/4], 19/16, [-5/2, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0061/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 62. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[3,5],"linear":[0,2],"point":[-1,1],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(5, 1/4, 2/5, 1, [[0, [-1, 1], 6, [-3, 7]], [1, [-1/4, -3/4], 0, [-3/4, -7/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(5, 1/4, 2/5, 1, [[0, [-1, 1], 6, [-3, 7]], [1, [-1/4, -3/4], 0, [-3/4, -7/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0062/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 63. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[1,2],"linear":[1,3],"point":[0,2],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(2, 1/4, 1, 1, [[0, [0, 2], 10, [1, 7]], [1, [-1/4, 1/4], 19/32, [3/4, 7/2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(2, 1/4, 1, 1, [[0, [0, 2], 10, [1, 7]], [1, [-1/4, 1/4], 19/32, [3/4, 7/2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0063/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 64. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[2,3],"linear":[2,-3],"point":[1,3],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(3, 1/4, 2/3, 1, [[0, [1, 3], 15/2, [4, 6]], [1, [0, 3/2], -9/8, [2, 3/2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(3, 1/4, 2/3, 1, [[0, [1, 3], 15/2, [4, 6]], [1, [0, 3/2], -9/8, [2, 3/2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0064/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 65. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[3,4],"linear":[-2,-2],"point":[2,-3],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(4, 1/4, 1/2, 1, [[0, [2, -3], 26, [4, -14]], [1, [1, 1/2], -1, [1, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(4, 1/4, 1/2, 1, [[0, [2, -3], 26, [4, -14]], [1, [1, 1/2], -1, [1, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0065/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 66. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[1,5],"linear":[-1,-1],"point":[-2,-2],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(5, 1/4, 2/5, 1, [[0, [-2, -2], 16, [-3, -11]], [1, [-5/4, 3/4], 43/16, [-9/4, 11/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(5, 1/4, 2/5, 1, [[0, [-2, -2], 16, [-3, -11]], [1, [-5/4, 3/4], 43/16, [-9/4, 11/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0066/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 67. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[2,2],"linear":[0,0],"point":[-1,-1],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(2, 1/4, 1, 1, [[0, [-1, -1], 2, [-2, -2]], [1, [-1/2, -1/2], 1/2, [-1, -1]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(2, 1/4, 1, 1, [[0, [-1, -1], 2, [-2, -2]], [1, [-1/2, -1/2], 1/2, [-1, -1]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0067/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 68. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[3,3],"linear":[1,1],"point":[0,0],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(3, 1/4, 2/3, 1, [[0, [0, 0], 0, [1, 1]], [1, [-1/4, -1/4], -5/16, [1/4, 1/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(3, 1/4, 2/3, 1, [[0, [0, 0], 0, [1, 1]], [1, [-1/4, -1/4], -5/16, [1/4, 1/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0068/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 69. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[1,4],"linear":[2,2],"point":[1,1],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(4, 1/4, 1/2, 1, [[0, [1, 1], 13/2, [3, 6]], [1, [1/4, -1/2], 1/32, [9/4, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(4, 1/4, 1/2, 1, [[0, [1, 1], 13/2, [3, 6]], [1, [1/4, -1/2], 1/32, [9/4, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0069/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 70. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[2,5],"linear":[-2,3],"point":[2,2],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(5, 1/4, 2/5, 1, [[0, [2, 2], 16, [2, 13]], [1, [3/2, -5/4], -19/32, [1, -13/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(5, 1/4, 2/5, 1, [[0, [2, 2], 16, [2, 13]], [1, [3/2, -5/4], -19/32, [1, -13/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0070/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 71. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[3,2],"linear":[-1,-3],"point":[-2,3],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(3, 1/4, 2/3, 1, [[0, [-2, 3], 8, [-7, 3]], [1, [-1/4, 9/4], -43/32, [-7/4, 3/2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(3, 1/4, 2/3, 1, [[0, [-2, 3], 8, [-7, 3]], [1, [-1/4, 9/4], -43/32, [-7/4, 3/2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0071/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 72. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[1,3],"linear":[0,-2],"point":[-1,-3],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(3, 1/4, 2/3, 1, [[0, [-1, -3], 20, [-1, -11]], [1, [-3/4, -1/4], 7/8, [-3/4, -11/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(3, 1/4, 2/3, 1, [[0, [-1, -3], 20, [-1, -11]], [1, [-3/4, -1/4], 7/8, [-3/4, -11/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0072/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 73. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[2,4],"linear":[1,-1],"point":[0,-2],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(4, 1/4, 1/2, 1, [[0, [0, -2], 10, [1, -9]], [1, [-1/4, 1/4], -5/16, [1/2, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(4, 1/4, 1/2, 1, [[0, [0, -2], 10, [1, -9]], [1, [-1/4, 1/4], -5/16, [1/2, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0073/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 74. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[3,5],"linear":[2,0],"point":[1,-1],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(5, 1/4, 2/5, 1, [[0, [1, -1], 6, [5, -5]], [1, [-1/4, 1/4], -1/4, [5/4, 5/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(5, 1/4, 2/5, 1, [[0, [1, -1], 6, [5, -5]], [1, [-1/4, 1/4], -1/4, [5/4, 5/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0074/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 75. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[1,2],"linear":[-2,1],"point":[2,0],"step_size":[1,4],"steps":1}；請依 [largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope] 回答。

safe範圍嚴格0<alpha<2/L，records包含initial point和每一步後狀態；有限nonincrease不是未驗證無限序列的convergence證書。

查看答案與解釋(2, 1/4, 1, 1, [[0, [2, 0], -2, [0, 1]], [1, [2, -1/4], -35/16, [0, 1/2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

Wave6第十三層核對=(2, 1/4, 1, 1, [[0, [2, 0], -2, [0, 1]], [1, [2, -1/4], -35/16, [0, 1/2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0075/)

梯度下降法 · qf-mod50-gd-safe-quadratic-trace · candidate





## 76. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"oversized_step","model":{"curvature":1,"point":1,"step_size":3}}；請依 [mode, curvature, point, step_size, next_point, objective_before, objective_after, automatic_descent_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(oversized_step, 1, 1, 3, -2, 1/2, 2, 否, positive_but_oversized_step_can_increase_the_objective)

Wave6第十三層核對=(oversized_step, 1, 1, 3, -2, 1/2, 2, 否, positive_but_oversized_step_can_increase_the_objective)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0076/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 77. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"zero_gradient_nonminimum","model":{"scale":2}}；請依 [mode, objective_description, stationary_point, gradient_statement, actual_point_status, minimum_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(zero_gradient_nonminimum, f(x)=-2*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)

Wave6第十三層核對=(zero_gradient_nonminimum, f(x)=-2*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0077/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 78. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"finite_trace_only","model":{"diagonal":[1,2],"linear":[2,-2],"point":[3,1],"step_size":[1,4],"steps":1}}；請依 [mode, step_count, iterate_objective_records, convergence_or_global_optimality_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(finite_trace_only, 1, [[0, [3, 1], 19/2], [1, [7/4, 1], 129/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)

Wave6第十三層核對=(finite_trace_only, 1, [[0, [3, 1], 19/2], [1, [7/4, 1], 129/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0078/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 79. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"constrained_confusion","model":{"gradient":4,"lower":-1,"point":0,"step_size":2,"upper":1}}；請依 [mode, lower_bound, upper_bound, feasible_point, unconstrained_next_point, constrained_descent_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(constrained_confusion, -1, 1, 0, -8, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)

Wave6第十三層核對=(constrained_confusion, -1, 1, 0, -8, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0079/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 80. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"oversized_step","model":{"curvature":5,"point":5,"step_size":3}}；請依 [mode, curvature, point, step_size, next_point, objective_before, objective_after, automatic_descent_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(oversized_step, 5, 5, 3, -70, 125/2, 12250, 否, positive_but_oversized_step_can_increase_the_objective)

Wave6第十三層核對=(oversized_step, 5, 5, 3, -70, 125/2, 12250, 否, positive_but_oversized_step_can_increase_the_objective)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0080/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 81. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"zero_gradient_nonminimum","model":{"scale":6}}；請依 [mode, objective_description, stationary_point, gradient_statement, actual_point_status, minimum_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(zero_gradient_nonminimum, f(x)=-6*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)

Wave6第十三層核對=(zero_gradient_nonminimum, f(x)=-6*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0081/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 82. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"finite_trace_only","model":{"diagonal":[1,2],"linear":[6,-6],"point":[7,1],"step_size":[1,4],"steps":1}}；請依 [mode, step_count, iterate_objective_records, convergence_or_global_optimality_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(finite_trace_only, 1, [[0, [7, 1], 123/2], [1, [15/4, 2], 689/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)

Wave6第十三層核對=(finite_trace_only, 1, [[0, [7, 1], 123/2], [1, [15/4, 2], 689/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0082/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 83. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"constrained_confusion","model":{"gradient":8,"lower":-1,"point":0,"step_size":2,"upper":1}}；請依 [mode, lower_bound, upper_bound, feasible_point, unconstrained_next_point, constrained_descent_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(constrained_confusion, -1, 1, 0, -16, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)

Wave6第十三層核對=(constrained_confusion, -1, 1, 0, -16, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0083/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 84. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"oversized_step","model":{"curvature":4,"point":9,"step_size":3}}；請依 [mode, curvature, point, step_size, next_point, objective_before, objective_after, automatic_descent_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(oversized_step, 4, 9, 3, -99, 162, 19602, 否, positive_but_oversized_step_can_increase_the_objective)

Wave6第十三層核對=(oversized_step, 4, 9, 3, -99, 162, 19602, 否, positive_but_oversized_step_can_increase_the_objective)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0084/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 85. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"zero_gradient_nonminimum","model":{"scale":10}}；請依 [mode, objective_description, stationary_point, gradient_statement, actual_point_status, minimum_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(zero_gradient_nonminimum, f(x)=-10*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)

Wave6第十三層核對=(zero_gradient_nonminimum, f(x)=-10*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0085/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 86. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"finite_trace_only","model":{"diagonal":[1,2],"linear":[10,-10],"point":[11,1],"step_size":[1,4],"steps":1}}；請依 [mode, step_count, iterate_objective_records, convergence_or_global_optimality_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(finite_trace_only, 1, [[0, [11, 1], 323/2], [1, [23/4, 3], 1697/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)

Wave6第十三層核對=(finite_trace_only, 1, [[0, [11, 1], 323/2], [1, [23/4, 3], 1697/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0086/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 87. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"constrained_confusion","model":{"gradient":12,"lower":-1,"point":0,"step_size":2,"upper":1}}；請依 [mode, lower_bound, upper_bound, feasible_point, unconstrained_next_point, constrained_descent_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(constrained_confusion, -1, 1, 0, -24, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)

Wave6第十三層核對=(constrained_confusion, -1, 1, 0, -24, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0087/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 88. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"oversized_step","model":{"curvature":3,"point":13,"step_size":3}}；請依 [mode, curvature, point, step_size, next_point, objective_before, objective_after, automatic_descent_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(oversized_step, 3, 13, 3, -104, 507/2, 16224, 否, positive_but_oversized_step_can_increase_the_objective)

Wave6第十三層核對=(oversized_step, 3, 13, 3, -104, 507/2, 16224, 否, positive_but_oversized_step_can_increase_the_objective)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0088/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 89. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"zero_gradient_nonminimum","model":{"scale":14}}；請依 [mode, objective_description, stationary_point, gradient_statement, actual_point_status, minimum_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(zero_gradient_nonminimum, f(x)=-14*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)

Wave6第十三層核對=(zero_gradient_nonminimum, f(x)=-14*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0089/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 90. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"finite_trace_only","model":{"diagonal":[1,2],"linear":[14,-14],"point":[15,1],"step_size":[1,4],"steps":1}}；請依 [mode, step_count, iterate_objective_records, convergence_or_global_optimality_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(finite_trace_only, 1, [[0, [15, 1], 619/2], [1, [31/4, 4], 3153/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)

Wave6第十三層核對=(finite_trace_only, 1, [[0, [15, 1], 619/2], [1, [31/4, 4], 3153/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0090/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 91. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"constrained_confusion","model":{"gradient":16,"lower":-1,"point":0,"step_size":2,"upper":1}}；請依 [mode, lower_bound, upper_bound, feasible_point, unconstrained_next_point, constrained_descent_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(constrained_confusion, -1, 1, 0, -32, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)

Wave6第十三層核對=(constrained_confusion, -1, 1, 0, -32, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0091/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 92. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"oversized_step","model":{"curvature":2,"point":17,"step_size":3}}；請依 [mode, curvature, point, step_size, next_point, objective_before, objective_after, automatic_descent_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(oversized_step, 2, 17, 3, -85, 289, 7225, 否, positive_but_oversized_step_can_increase_the_objective)

Wave6第十三層核對=(oversized_step, 2, 17, 3, -85, 289, 7225, 否, positive_but_oversized_step_can_increase_the_objective)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0092/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 93. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"zero_gradient_nonminimum","model":{"scale":18}}；請依 [mode, objective_description, stationary_point, gradient_statement, actual_point_status, minimum_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(zero_gradient_nonminimum, f(x)=-18*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)

Wave6第十三層核對=(zero_gradient_nonminimum, f(x)=-18*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0093/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 94. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"finite_trace_only","model":{"diagonal":[1,2],"linear":[18,-18],"point":[19,1],"step_size":[1,4],"steps":1}}；請依 [mode, step_count, iterate_objective_records, convergence_or_global_optimality_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(finite_trace_only, 1, [[0, [19, 1], 1011/2], [1, [39/4, 5], 5057/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)

Wave6第十三層核對=(finite_trace_only, 1, [[0, [19, 1], 1011/2], [1, [39/4, 5], 5057/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0094/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 95. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"constrained_confusion","model":{"gradient":20,"lower":-1,"point":0,"step_size":2,"upper":1}}；請依 [mode, lower_bound, upper_bound, feasible_point, unconstrained_next_point, constrained_descent_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(constrained_confusion, -1, 1, 0, -40, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)

Wave6第十三層核對=(constrained_confusion, -1, 1, 0, -40, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0095/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 96. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"oversized_step","model":{"curvature":1,"point":21,"step_size":3}}；請依 [mode, curvature, point, step_size, next_point, objective_before, objective_after, automatic_descent_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(oversized_step, 1, 21, 3, -42, 441/2, 882, 否, positive_but_oversized_step_can_increase_the_objective)

Wave6第十三層核對=(oversized_step, 1, 21, 3, -42, 441/2, 882, 否, positive_but_oversized_step_can_increase_the_objective)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0096/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 97. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"zero_gradient_nonminimum","model":{"scale":22}}；請依 [mode, objective_description, stationary_point, gradient_statement, actual_point_status, minimum_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(zero_gradient_nonminimum, f(x)=-22*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)

Wave6第十三層核對=(zero_gradient_nonminimum, f(x)=-22*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0097/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 98. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"finite_trace_only","model":{"diagonal":[1,2],"linear":[22,-22],"point":[23,1],"step_size":[1,4],"steps":1}}；請依 [mode, step_count, iterate_objective_records, convergence_or_global_optimality_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(finite_trace_only, 1, [[0, [23, 1], 1499/2], [1, [47/4, 6], 7409/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)

Wave6第十三層核對=(finite_trace_only, 1, [[0, [23, 1], 1499/2], [1, [47/4, 6], 7409/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0098/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 99. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"constrained_confusion","model":{"gradient":24,"lower":-1,"point":0,"step_size":2,"upper":1}}；請依 [mode, lower_bound, upper_bound, feasible_point, unconstrained_next_point, constrained_descent_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(constrained_confusion, -1, 1, 0, -48, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)

Wave6第十三層核對=(constrained_confusion, -1, 1, 0, -48, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0099/)

梯度下降法 · qf-mod50-gd-boundary · candidate





## 100. 梯度下降

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={"mode":"oversized_step","model":{"curvature":5,"point":25,"step_size":3}}；請依 [mode, curvature, point, step_size, next_point, objective_before, objective_after, automatic_descent_claim, boundary] 回答。

四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。

查看答案與解釋(oversized_step, 5, 25, 3, -350, 3125/2, 306250, 否, positive_but_oversized_step_can_increase_the_objective)

Wave6第十三層核對=(oversized_step, 5, 25, 3, -350, 3125/2, 306250, 否, positive_but_oversized_step_can_increase_the_objective)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r03-0100/)

梯度下降法 · qf-mod50-gd-boundary · candidate



[原始題目 JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261119-r03-gradient-descent-foundations/questions.jsonl) · [驗收報告](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261119-r03-gradient-descent-foundations/validation-report.json)