{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261025-r03-convex-optimization-problem-0094", "pilot_number": "R0022-11969", "number": 11969, "family_id": "qf-mod25-convexopt-pairwise-feasible-comparison", "title": { "zh-Hant": "第 11969 題：凸最佳化問題", "en": "Question 11969: Compare Only Two Eligible Candidates, Not the Whole Problem" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261025-r03-convex-optimization-problem-0094", "batch_id": "batch-20261013-advance-20261025-r03-convex-optimization-problem-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-convex-optimization-problem" ], "title_zh": "凸最佳化問題", "stem_zh": "凸目標資格與凸可行域資格分開。操作=convexopt-pairwise-feasible-comparison；資料={\"coefficients\":[2,6,1],\"interval\":[-20,21],\"left\":[1,1],\"right\":[1,1]}。請依 [coefficients,interval,left,left_feasible,left_value,right,right_feasible,right_value,relation,scope] 欄序回答。", "answer": { "kind": "expression", "value": "([2, 6, 1], [-20, 21], 1, 是, 9, 1, 是, 9, equal, pairwise_only_no_global_optimum_claim)", "explanation_zh": "收束波核對=([2, 6, 1], [-20, 21], 1, 是, 9, 1, 是, 9, equal, pairwise_only_no_global_optimum_claim)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只核可行性求值或兩候選比較；不宣稱通用求解器全域最優存在唯一KKT或收斂。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "凸最佳化問題" ] }, "variant": { "family_id": "qf-mod25-convexopt-pairwise-feasible-comparison", "seed": 202610253094, "parameters": { "coefficients": [ 2, 6, 1 ], "interval": [ -20, 21 ], "left": [ 1, 1 ], "right": [ 1, 1 ], "domain": "Exact_interval_convex_optimization", "operation": "convexopt-pairwise-feasible-comparison", "semantic_template": "qf-mod25-convexopt-pairwise-feasible-comparison/{\"coefficients\":[2,6,1],\"interval\":[-20,21],\"left\":[1,1],\"right\":[1,1]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-13", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261013-advance-20261025-r03-convex-optimization-problem-0094", "batch_id": "batch-20261013-advance-20261025-r03-convex-optimization-problem-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "95dc8a40af78b4165c92ffb903cfc9d714904fc878389dcf817957982095f118", "family_id": "qf-mod25-convexopt-pairwise-feasible-comparison", "seed": 202610253094, "semantic_template": "qf-mod25-convexopt-pairwise-feasible-comparison/{\"coefficients\":[2,6,1],\"interval\":[-20,21],\"left\":[1,1],\"right\":[1,1]}" }, "fields": { "title": "Compare Only Two Eligible Candidates, Not the Whole Problem", "stem": "Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 6, 1], not ascending-power order. The declared interval [lower,upper]=[-20, 21] belongs to the function data. left=1/1, right=1/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.", "answer": { "kind": "expression", "value": "([2, 6, 1], [-20, 21], 1, Yes, 9, 1, Yes, 9, equal, pairwise_only_no_global_optimum_claim)", "explanation": "Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 6, 1], [-20, 21], 1, Yes, 9, 1, Yes, 9, equal, pairwise_only_no_global_optimum_claim)." }, "skills": [ "Compare Only Two Eligible Candidates, Not the Whole Problem" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "凸目標資格與凸可行域資格分開。操作=convexopt-pairwise-feasible-comparison；資料={\"coefficients\":[2,6,1],\"interval\":[-20,21],\"left\":[1,1],\"right\":[1,1]}。請依 [coefficients,interval,left,left_feasible,left_value,right,right_feasible,right_value,relation,scope] 欄序回答。", "en": "Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 6, 1], not ascending-power order. The declared interval [lower,upper]=[-20, 21] belongs to the function data. left=1/1, right=1/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "declared interval record=[-20, 21]", "after": "ordered_nonempty=是;nondegenerate=是", "relation": "implies", "rule_id": "interval_certificate_qualification_not_empty_set_nonconvexity", "conditions": [], "explanation": { "zh-Hant": "此非空區間協定用lower≤upper，非退化證書另需lower