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
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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。
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([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）
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。
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([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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
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）
線性規劃對偶 · qf-mod46-lpdual-strong-boundary · candidate
原始題目 JSONL · 驗收報告