# 線性規劃對偶基礎 · 脈衝 R04

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-20261115-r04-linear-program-duality-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. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[2,3],"c":[1,1]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [2, 3], [1, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [2, 3], [1, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0001/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 2. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[3,4],"c":[2,2]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [3, 4], [2, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [3, 4], [2, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0002/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 3. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[4,5],"c":[3,3]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [4, 5], [3, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [4, 5], [3, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0003/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 4. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[5,6],"c":[4,4]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [5, 6], [4, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [5, 6], [4, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0004/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 5. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[6,7],"c":[1,5]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [6, 7], [1, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [6, 7], [1, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0005/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 6. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[7,8],"c":[2,1]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [7, 8], [2, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [7, 8], [2, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0006/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 7. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[8,9],"c":[3,2]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [8, 9], [3, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [8, 9], [3, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0007/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 8. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[9,10],"c":[4,3]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [9, 10], [4, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [9, 10], [4, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0008/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 9. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[10,11],"c":[1,4]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [10, 11], [1, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [10, 11], [1, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0009/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 10. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[11,12],"c":[2,5]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [11, 12], [2, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [11, 12], [2, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0010/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 11. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[12,13],"c":[3,1]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [12, 13], [3, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [12, 13], [3, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0011/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 12. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[13,14],"c":[4,2]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [13, 14], [4, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [13, 14], [4, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0012/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 13. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[14,15],"c":[1,3]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [14, 15], [1, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [14, 15], [1, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0013/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 14. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[15,16],"c":[2,4]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [15, 16], [2, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [15, 16], [2, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0014/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 15. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[16,17],"c":[3,5]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [16, 17], [3, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [16, 17], [3, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0015/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 16. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[17,18],"c":[4,1]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [17, 18], [4, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [17, 18], [4, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0016/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 17. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[18,19],"c":[1,2]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [18, 19], [1, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [18, 19], [1, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0017/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 18. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[19,20],"c":[2,3]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [19, 20], [2, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [19, 20], [2, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0018/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 19. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[20,21],"c":[3,4]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [20, 21], [3, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [20, 21], [3, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0019/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 20. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[21,22],"c":[4,5]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [21, 22], [4, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [21, 22], [4, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0020/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 21. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[22,23],"c":[1,1]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [22, 23], [1, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [22, 23], [1, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0021/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 22. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[23,24],"c":[2,2]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [23, 24], [2, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [23, 24], [2, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0022/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 23. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[24,25],"c":[3,3]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [24, 25], [3, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [24, 25], [3, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0023/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 24. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[25,26],"c":[4,4]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [25, 26], [4, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [25, 26], [4, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0024/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 25. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={"A":[[1,0],[0,1]],"b":[26,27],"c":[1,5]}。請依 [constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate] 欄序回答。

查看答案與解釋(2, 2, [[1, 0], [0, 1]], [26, 27], [1, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)

Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [26, 27], [1, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0025/)

線性規劃對偶 · qf-mod46-lpdual-construction-qualification · candidate





## 26. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[2,3],"c":[1,1],"x":[[2,2],[3,2]],"y":[2,2]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([1, 3/2], [2, 2], [1, 3/2], [2, 2], 是, 是, 5/2, 5, 10, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([1, 3/2], [2, 2], [1, 3/2], [2, 2], 是, 是, 5/2, 5, 10, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0026/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 27. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[3,4],"c":[2,2],"x":[[3,2],[4,2]],"y":[3,3]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([3/2, 2], [3, 3], [3/2, 2], [3, 3], 是, 是, 7, 21/2, 21, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([3/2, 2], [3, 3], [3/2, 2], [3, 3], 是, 是, 7, 21/2, 21, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0027/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 28. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[4,5],"c":[3,3],"x":[[4,2],[5,2]],"y":[4,4]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([2, 5/2], [4, 4], [2, 5/2], [4, 4], 是, 是, 27/2, 18, 36, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([2, 5/2], [4, 4], [2, 5/2], [4, 4], 是, 是, 27/2, 18, 36, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0028/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 29. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[5,6],"c":[4,4],"x":[[5,2],[6,2]],"y":[5,5]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([5/2, 3], [5, 5], [5/2, 3], [5, 5], 是, 是, 22, 55/2, 55, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([5/2, 3], [5, 5], [5/2, 3], [5, 5], 是, 是, 22, 55/2, 55, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0029/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 30. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[6,7],"c":[1,5],"x":[[6,2],[7,2]],"y":[2,6]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([3, 7/2], [2, 6], [3, 7/2], [2, 6], 是, 是, 41/2, 27, 54, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([3, 7/2], [2, 6], [3, 7/2], [2, 6], 是, 是, 41/2, 27, 54, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0030/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 31. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[7,8],"c":[2,1],"x":[[7,2],[8,2]],"y":[3,2]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([7/2, 4], [3, 2], [7/2, 4], [3, 2], 是, 是, 11, 37/2, 37, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([7/2, 4], [3, 2], [7/2, 4], [3, 2], 是, 是, 11, 37/2, 37, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0031/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 32. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[8,9],"c":[3,2],"x":[[8,2],[9,2]],"y":[4,3]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([4, 9/2], [4, 3], [4, 9/2], [4, 3], 是, 是, 21, 59/2, 59, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([4, 9/2], [4, 3], [4, 9/2], [4, 3], 是, 是, 21, 59/2, 59, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0032/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 33. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[9,10],"c":[4,3],"x":[[9,2],[10,2]],"y":[5,4]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([9/2, 5], [5, 4], [9/2, 5], [5, 4], 是, 是, 33, 85/2, 85, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([9/2, 5], [5, 4], [9/2, 5], [5, 4], 是, 是, 33, 85/2, 85, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0033/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 34. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[10,11],"c":[1,4],"x":[[10,2],[11,2]],"y":[2,5]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([5, 11/2], [2, 5], [5, 11/2], [2, 5], 是, 是, 27, 75/2, 75, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([5, 11/2], [2, 5], [5, 11/2], [2, 5], 是, 是, 27, 75/2, 75, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0034/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 35. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[11,12],"c":[2,5],"x":[[11,2],[12,2]],"y":[3,6]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([11/2, 6], [3, 6], [11/2, 6], [3, 6], 是, 是, 41, 105/2, 105, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([11/2, 6], [3, 6], [11/2, 6], [3, 6], 是, 是, 41, 105/2, 105, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0035/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 36. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[12,13],"c":[3,1],"x":[[12,2],[13,2]],"y":[4,2]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([6, 13/2], [4, 2], [6, 13/2], [4, 2], 是, 是, 49/2, 37, 74, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([6, 13/2], [4, 2], [6, 13/2], [4, 2], 是, 是, 49/2, 37, 74, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0036/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 37. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[13,14],"c":[4,2],"x":[[13,2],[14,2]],"y":[5,3]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([13/2, 7], [5, 3], [13/2, 7], [5, 3], 是, 是, 40, 107/2, 107, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([13/2, 7], [5, 3], [13/2, 7], [5, 3], 是, 是, 40, 107/2, 107, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0037/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 38. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[14,15],"c":[1,3],"x":[[14,2],[15,2]],"y":[2,4]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([7, 15/2], [2, 4], [7, 15/2], [2, 4], 是, 是, 59/2, 44, 88, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([7, 15/2], [2, 4], [7, 15/2], [2, 4], 是, 是, 59/2, 44, 88, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0038/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 39. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[15,16],"c":[2,4],"x":[[15,2],[16,2]],"y":[3,5]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([15/2, 8], [3, 5], [15/2, 8], [3, 5], 是, 是, 47, 125/2, 125, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([15/2, 8], [3, 5], [15/2, 8], [3, 5], 是, 是, 47, 125/2, 125, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0039/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 40. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[16,17],"c":[3,5],"x":[[16,2],[17,2]],"y":[4,6]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([8, 17/2], [4, 6], [8, 17/2], [4, 6], 是, 是, 133/2, 83, 166, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([8, 17/2], [4, 6], [8, 17/2], [4, 6], 是, 是, 133/2, 83, 166, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0040/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 41. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[17,18],"c":[4,1],"x":[[17,2],[18,2]],"y":[5,2]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([17/2, 9], [5, 2], [17/2, 9], [5, 2], 是, 是, 43, 121/2, 121, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([17/2, 9], [5, 2], [17/2, 9], [5, 2], 是, 是, 43, 121/2, 121, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0041/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 42. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[18,19],"c":[1,2],"x":[[18,2],[19,2]],"y":[2,3]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([9, 19/2], [2, 3], [9, 19/2], [2, 3], 是, 是, 28, 93/2, 93, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([9, 19/2], [2, 3], [9, 19/2], [2, 3], 是, 是, 28, 93/2, 93, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0042/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 43. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[19,20],"c":[2,3],"x":[[19,2],[20,2]],"y":[3,4]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([19/2, 10], [3, 4], [19/2, 10], [3, 4], 是, 是, 49, 137/2, 137, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([19/2, 10], [3, 4], [19/2, 10], [3, 4], 是, 是, 49, 137/2, 137, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0043/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 44. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[20,21],"c":[3,4],"x":[[20,2],[21,2]],"y":[4,5]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([10, 21/2], [4, 5], [10, 21/2], [4, 5], 是, 是, 72, 185/2, 185, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([10, 21/2], [4, 5], [10, 21/2], [4, 5], 是, 是, 72, 185/2, 185, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0044/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 45. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[21,22],"c":[4,5],"x":[[21,2],[22,2]],"y":[5,6]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([21/2, 11], [5, 6], [21/2, 11], [5, 6], 是, 是, 97, 237/2, 237, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([21/2, 11], [5, 6], [21/2, 11], [5, 6], 是, 是, 97, 237/2, 237, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0045/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 46. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[22,23],"c":[1,1],"x":[[22,2],[23,2]],"y":[2,2]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([11, 23/2], [2, 2], [11, 23/2], [2, 2], 是, 是, 45/2, 45, 90, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([11, 23/2], [2, 2], [11, 23/2], [2, 2], 是, 是, 45/2, 45, 90, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0046/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 47. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[23,24],"c":[2,2],"x":[[23,2],[24,2]],"y":[3,3]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([23/2, 12], [3, 3], [23/2, 12], [3, 3], 是, 是, 47, 141/2, 141, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([23/2, 12], [3, 3], [23/2, 12], [3, 3], 是, 是, 47, 141/2, 141, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0047/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 48. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[24,25],"c":[3,3],"x":[[24,2],[25,2]],"y":[4,4]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([12, 25/2], [4, 4], [12, 25/2], [4, 4], 是, 是, 147/2, 98, 196, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([12, 25/2], [4, 4], [12, 25/2], [4, 4], 是, 是, 147/2, 98, 196, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0048/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 49. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[25,26],"c":[4,4],"x":[[25,2],[26,2]],"y":[5,5]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([25/2, 13], [5, 5], [25/2, 13], [5, 5], 是, 是, 102, 255/2, 255, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([25/2, 13], [5, 5], [25/2, 13], [5, 5], 是, 是, 102, 255/2, 255, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0049/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 50. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={"A":[[1,0],[0,1]],"b":[26,27],"c":[1,5],"x":[[26,2],[27,2]],"y":[2,6]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate] 欄序回答。

查看答案與解釋([13, 27/2], [2, 6], [13, 27/2], [2, 6], 是, 是, 161/2, 107, 214, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

Wave6第九層核對=([13, 27/2], [2, 6], [13, 27/2], [2, 6], 是, 是, 161/2, 107, 214, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0050/)

線性規劃對偶 · qf-mod46-lpdual-weak-duality-chain · candidate





## 51. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[2,3],"c":[1,1],"x":[2,3],"y":[1,1]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([2, 3], [1, 1], [2, 3], [1, 1], 是, 是, 5, 5, 5, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([2, 3], [1, 1], [2, 3], [1, 1], 是, 是, 5, 5, 5, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0051/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 52. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[3,4],"c":[2,2],"x":[3,4],"y":[2,2]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([3, 4], [2, 2], [3, 4], [2, 2], 是, 是, 14, 14, 14, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([3, 4], [2, 2], [3, 4], [2, 2], 是, 是, 14, 14, 14, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0052/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 53. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[4,5],"c":[3,3],"x":[4,5],"y":[3,3]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([4, 5], [3, 3], [4, 5], [3, 3], 是, 是, 27, 27, 27, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([4, 5], [3, 3], [4, 5], [3, 3], 是, 是, 27, 27, 27, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0053/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 54. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[5,6],"c":[4,4],"x":[5,6],"y":[4,4]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([5, 6], [4, 4], [5, 6], [4, 4], 是, 是, 44, 44, 44, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([5, 6], [4, 4], [5, 6], [4, 4], 是, 是, 44, 44, 44, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0054/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 55. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[6,7],"c":[1,5],"x":[6,7],"y":[1,5]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([6, 7], [1, 5], [6, 7], [1, 5], 是, 是, 41, 41, 41, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([6, 7], [1, 5], [6, 7], [1, 5], 是, 是, 41, 41, 41, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0055/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 56. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[7,8],"c":[2,1],"x":[7,8],"y":[2,1]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([7, 8], [2, 1], [7, 8], [2, 1], 是, 是, 22, 22, 22, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([7, 8], [2, 1], [7, 8], [2, 1], 是, 是, 22, 22, 22, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0056/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 57. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[8,9],"c":[3,2],"x":[8,9],"y":[3,2]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([8, 9], [3, 2], [8, 9], [3, 2], 是, 是, 42, 42, 42, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([8, 9], [3, 2], [8, 9], [3, 2], 是, 是, 42, 42, 42, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0057/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 58. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[9,10],"c":[4,3],"x":[9,10],"y":[4,3]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([9, 10], [4, 3], [9, 10], [4, 3], 是, 是, 66, 66, 66, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([9, 10], [4, 3], [9, 10], [4, 3], 是, 是, 66, 66, 66, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0058/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 59. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[10,11],"c":[1,4],"x":[10,11],"y":[1,4]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([10, 11], [1, 4], [10, 11], [1, 4], 是, 是, 54, 54, 54, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([10, 11], [1, 4], [10, 11], [1, 4], 是, 是, 54, 54, 54, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0059/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 60. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[11,12],"c":[2,5],"x":[11,12],"y":[2,5]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([11, 12], [2, 5], [11, 12], [2, 5], 是, 是, 82, 82, 82, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([11, 12], [2, 5], [11, 12], [2, 5], 是, 是, 82, 82, 82, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0060/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 61. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[12,13],"c":[3,1],"x":[12,13],"y":[3,1]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([12, 13], [3, 1], [12, 13], [3, 1], 是, 是, 49, 49, 49, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([12, 13], [3, 1], [12, 13], [3, 1], 是, 是, 49, 49, 49, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0061/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 62. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[13,14],"c":[4,2],"x":[13,14],"y":[4,2]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([13, 14], [4, 2], [13, 14], [4, 2], 是, 是, 80, 80, 80, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([13, 14], [4, 2], [13, 14], [4, 2], 是, 是, 80, 80, 80, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0062/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 63. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[14,15],"c":[1,3],"x":[14,15],"y":[1,3]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([14, 15], [1, 3], [14, 15], [1, 3], 是, 是, 59, 59, 59, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([14, 15], [1, 3], [14, 15], [1, 3], 是, 是, 59, 59, 59, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0063/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 64. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[15,16],"c":[2,4],"x":[15,16],"y":[2,4]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([15, 16], [2, 4], [15, 16], [2, 4], 是, 是, 94, 94, 94, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([15, 16], [2, 4], [15, 16], [2, 4], 是, 是, 94, 94, 94, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0064/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 65. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[16,17],"c":[3,5],"x":[16,17],"y":[3,5]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([16, 17], [3, 5], [16, 17], [3, 5], 是, 是, 133, 133, 133, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([16, 17], [3, 5], [16, 17], [3, 5], 是, 是, 133, 133, 133, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0065/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 66. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[17,18],"c":[4,1],"x":[17,18],"y":[4,1]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([17, 18], [4, 1], [17, 18], [4, 1], 是, 是, 86, 86, 86, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([17, 18], [4, 1], [17, 18], [4, 1], 是, 是, 86, 86, 86, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0066/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 67. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[18,19],"c":[1,2],"x":[18,19],"y":[1,2]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([18, 19], [1, 2], [18, 19], [1, 2], 是, 是, 56, 56, 56, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([18, 19], [1, 2], [18, 19], [1, 2], 是, 是, 56, 56, 56, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0067/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 68. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[19,20],"c":[2,3],"x":[19,20],"y":[2,3]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([19, 20], [2, 3], [19, 20], [2, 3], 是, 是, 98, 98, 98, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([19, 20], [2, 3], [19, 20], [2, 3], 是, 是, 98, 98, 98, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0068/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 69. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[20,21],"c":[3,4],"x":[20,21],"y":[3,4]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([20, 21], [3, 4], [20, 21], [3, 4], 是, 是, 144, 144, 144, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([20, 21], [3, 4], [20, 21], [3, 4], 是, 是, 144, 144, 144, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0069/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 70. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[21,22],"c":[4,5],"x":[21,22],"y":[4,5]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([21, 22], [4, 5], [21, 22], [4, 5], 是, 是, 194, 194, 194, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([21, 22], [4, 5], [21, 22], [4, 5], 是, 是, 194, 194, 194, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0070/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 71. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[22,23],"c":[1,1],"x":[22,23],"y":[1,1]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([22, 23], [1, 1], [22, 23], [1, 1], 是, 是, 45, 45, 45, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([22, 23], [1, 1], [22, 23], [1, 1], 是, 是, 45, 45, 45, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0071/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 72. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[23,24],"c":[2,2],"x":[23,24],"y":[2,2]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([23, 24], [2, 2], [23, 24], [2, 2], 是, 是, 94, 94, 94, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([23, 24], [2, 2], [23, 24], [2, 2], 是, 是, 94, 94, 94, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0072/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 73. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[24,25],"c":[3,3],"x":[24,25],"y":[3,3]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([24, 25], [3, 3], [24, 25], [3, 3], 是, 是, 147, 147, 147, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([24, 25], [3, 3], [24, 25], [3, 3], 是, 是, 147, 147, 147, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0073/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 74. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[25,26],"c":[4,4],"x":[25,26],"y":[4,4]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([25, 26], [4, 4], [25, 26], [4, 4], 是, 是, 204, 204, 204, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([25, 26], [4, 4], [25, 26], [4, 4], 是, 是, 204, 204, 204, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0074/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 75. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={"A":[[1,0],[0,1]],"b":[26,27],"c":[1,5],"x":[26,27],"y":[1,5]}。請依 [x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate] 欄序回答。

查看答案與解釋([26, 27], [1, 5], [26, 27], [1, 5], 是, 是, 161, 161, 161, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)

Wave6第九層核對=([26, 27], [1, 5], [26, 27], [1, 5], 是, 是, 161, 161, 161, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0075/)

線性規劃對偶 · qf-mod46-lpdual-zero-gap-certificate · candidate





## 76. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"primal_infeasible_equal_objective","A":[[1]],"b":[2],"c":[1],"x":[3],"y":[[3,2]]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(primal_infeasible_equal_objective, [3], [3/2], [3], [3/2], 否, 是, 3, 9/2, 3, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(primal_infeasible_equal_objective, [3], [3/2], [3], [3/2], 否, 是, 3, 9/2, 3, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0076/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 77. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"dual_infeasible_equal_objective","A":[[1]],"b":[1],"c":[3],"x":[[1,3]],"y":[1]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(dual_infeasible_equal_objective, [1/3], [1], [1/3], [1], 是, 否, 1, 1/3, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(dual_infeasible_equal_objective, [1/3], [1], [1/3], [1], 是, 否, 1, 1/3, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0077/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 78. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"finite_grid_not_general","A":[[1]],"b":[4],"c":[1],"x":[1],"y":[2]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 8, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 8, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0078/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 79. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"primal_infeasible_equal_objective","A":[[1]],"b":[5],"c":[1],"x":[6],"y":[[6,5]]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(primal_infeasible_equal_objective, [6], [6/5], [6], [6/5], 否, 是, 6, 36/5, 6, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(primal_infeasible_equal_objective, [6], [6/5], [6], [6/5], 否, 是, 6, 36/5, 6, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0079/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 80. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"dual_infeasible_equal_objective","A":[[1]],"b":[1],"c":[6],"x":[[1,6]],"y":[1]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(dual_infeasible_equal_objective, [1/6], [1], [1/6], [1], 是, 否, 1, 1/6, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(dual_infeasible_equal_objective, [1/6], [1], [1/6], [1], 是, 否, 1, 1/6, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0080/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 81. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"finite_grid_not_general","A":[[1]],"b":[7],"c":[1],"x":[1],"y":[2]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 14, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 14, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0081/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 82. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"primal_infeasible_equal_objective","A":[[1]],"b":[8],"c":[1],"x":[9],"y":[[9,8]]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(primal_infeasible_equal_objective, [9], [9/8], [9], [9/8], 否, 是, 9, 81/8, 9, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(primal_infeasible_equal_objective, [9], [9/8], [9], [9/8], 否, 是, 9, 81/8, 9, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0082/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 83. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"dual_infeasible_equal_objective","A":[[1]],"b":[1],"c":[9],"x":[[1,9]],"y":[1]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(dual_infeasible_equal_objective, [1/9], [1], [1/9], [1], 是, 否, 1, 1/9, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(dual_infeasible_equal_objective, [1/9], [1], [1/9], [1], 是, 否, 1, 1/9, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0083/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 84. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"finite_grid_not_general","A":[[1]],"b":[10],"c":[1],"x":[1],"y":[2]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 20, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 20, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0084/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 85. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"primal_infeasible_equal_objective","A":[[1]],"b":[11],"c":[1],"x":[12],"y":[[12,11]]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(primal_infeasible_equal_objective, [12], [12/11], [12], [12/11], 否, 是, 12, 144/11, 12, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(primal_infeasible_equal_objective, [12], [12/11], [12], [12/11], 否, 是, 12, 144/11, 12, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0085/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 86. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"dual_infeasible_equal_objective","A":[[1]],"b":[1],"c":[12],"x":[[1,12]],"y":[1]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(dual_infeasible_equal_objective, [1/12], [1], [1/12], [1], 是, 否, 1, 1/12, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(dual_infeasible_equal_objective, [1/12], [1], [1/12], [1], 是, 否, 1, 1/12, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0086/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 87. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"finite_grid_not_general","A":[[1]],"b":[13],"c":[1],"x":[1],"y":[2]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 26, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 26, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0087/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 88. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"primal_infeasible_equal_objective","A":[[1]],"b":[14],"c":[1],"x":[15],"y":[[15,14]]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(primal_infeasible_equal_objective, [15], [15/14], [15], [15/14], 否, 是, 15, 225/14, 15, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(primal_infeasible_equal_objective, [15], [15/14], [15], [15/14], 否, 是, 15, 225/14, 15, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0088/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 89. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"dual_infeasible_equal_objective","A":[[1]],"b":[1],"c":[15],"x":[[1,15]],"y":[1]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(dual_infeasible_equal_objective, [1/15], [1], [1/15], [1], 是, 否, 1, 1/15, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(dual_infeasible_equal_objective, [1/15], [1], [1/15], [1], 是, 否, 1, 1/15, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0089/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 90. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"finite_grid_not_general","A":[[1]],"b":[16],"c":[1],"x":[1],"y":[2]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 32, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 32, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0090/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 91. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"primal_infeasible_equal_objective","A":[[1]],"b":[17],"c":[1],"x":[18],"y":[[18,17]]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(primal_infeasible_equal_objective, [18], [18/17], [18], [18/17], 否, 是, 18, 324/17, 18, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(primal_infeasible_equal_objective, [18], [18/17], [18], [18/17], 否, 是, 18, 324/17, 18, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0091/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 92. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"dual_infeasible_equal_objective","A":[[1]],"b":[1],"c":[18],"x":[[1,18]],"y":[1]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(dual_infeasible_equal_objective, [1/18], [1], [1/18], [1], 是, 否, 1, 1/18, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(dual_infeasible_equal_objective, [1/18], [1], [1/18], [1], 是, 否, 1, 1/18, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0092/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 93. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"finite_grid_not_general","A":[[1]],"b":[19],"c":[1],"x":[1],"y":[2]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 38, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 38, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0093/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 94. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"primal_infeasible_equal_objective","A":[[1]],"b":[20],"c":[1],"x":[21],"y":[[21,20]]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(primal_infeasible_equal_objective, [21], [21/20], [21], [21/20], 否, 是, 21, 441/20, 21, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(primal_infeasible_equal_objective, [21], [21/20], [21], [21/20], 否, 是, 21, 441/20, 21, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0094/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 95. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"dual_infeasible_equal_objective","A":[[1]],"b":[1],"c":[21],"x":[[1,21]],"y":[1]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(dual_infeasible_equal_objective, [1/21], [1], [1/21], [1], 是, 否, 1, 1/21, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(dual_infeasible_equal_objective, [1/21], [1], [1/21], [1], 是, 否, 1, 1/21, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0095/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 96. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"finite_grid_not_general","A":[[1]],"b":[22],"c":[1],"x":[1],"y":[2]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 44, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 44, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0096/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 97. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"primal_infeasible_equal_objective","A":[[1]],"b":[23],"c":[1],"x":[24],"y":[[24,23]]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(primal_infeasible_equal_objective, [24], [24/23], [24], [24/23], 否, 是, 24, 576/23, 24, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(primal_infeasible_equal_objective, [24], [24/23], [24], [24/23], 否, 是, 24, 576/23, 24, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0097/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 98. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"dual_infeasible_equal_objective","A":[[1]],"b":[1],"c":[24],"x":[[1,24]],"y":[1]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(dual_infeasible_equal_objective, [1/24], [1], [1/24], [1], 是, 否, 1, 1/24, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(dual_infeasible_equal_objective, [1/24], [1], [1/24], [1], 是, 否, 1, 1/24, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0098/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 99. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"finite_grid_not_general","A":[[1]],"b":[25],"c":[1],"x":[1],"y":[2]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 50, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 50, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0099/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate





## 100. 線性規劃對偶

固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={"mode":"primal_infeasible_equal_objective","A":[[1]],"b":[26],"c":[1],"x":[27],"y":[[27,26]]}。請依 [mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary] 欄序回答。

三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。

查看答案與解釋(primal_infeasible_equal_objective, [27], [27/26], [27], [27/26], 否, 是, 27, 729/26, 27, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)

Wave6第九層核對=(primal_infeasible_equal_objective, [27], [27/26], [27], [27/26], 否, 是, 27, 729/26, 27, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r04-linear-program-duality-0100/)

線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate



[原始題目 JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261115-r04-linear-program-duality-foundations/questions.jsonl) · [驗收報告](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261115-r04-linear-program-duality-foundations/validation-report.json)