{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261025-r03-convex-optimization-problem-0046", "pilot_number": "R0022-11921", "number": 11921, "family_id": "qf-mod25-convexopt-feasible-point-audit", "title": { "zh-Hant": "第 11921 題：凸最佳化問題", "en": "Question 11921: Feasibility Alone Supplies No Optimum" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261025-r03-convex-optimization-problem-0046", "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-feasible-point-audit；資料={\"interval\":[-22,23],\"point\":[3,1]}。請依 [interval,point,feasible,scope] 欄序回答。", "answer": { "kind": "expression", "value": "([-22, 23], 3, 是, feasibility_only)", "explanation_zh": "收束波核對=([-22, 23], 3, 是, feasibility_only)；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-feasible-point-audit", "seed": 202610253046, "parameters": { "interval": [ -22, 23 ], "point": [ 3, 1 ], "domain": "Exact_interval_convex_optimization", "operation": "convexopt-feasible-point-audit", "semantic_template": "qf-mod25-convexopt-feasible-point-audit/{\"interval\":[-22,23],\"point\":[3,1]}" } }, "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-0046", "batch_id": "batch-20261013-advance-20261025-r03-convex-optimization-problem-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "ca86dd0cd7a348ae1471bde0acd8240c871dd2bbbbcf05a3fd18455879ba0408", "family_id": "qf-mod25-convexopt-feasible-point-audit", "seed": 202610253046, "semantic_template": "qf-mod25-convexopt-feasible-point-audit/{\"interval\":[-22,23],\"point\":[3,1]}" }, "fields": { "title": "Feasibility Alone Supplies No Optimum", "stem": "Declared nonempty interval record=[-22, 23], candidate=3/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. 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": "([-22, 23], 3, Yes, feasibility_only)", "explanation": "Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-22, 23], 3, Yes, feasibility_only)." }, "skills": [ "Feasibility Alone Supplies No Optimum" ], "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-feasible-point-audit；資料={\"interval\":[-22,23],\"point\":[3,1]}。請依 [interval,point,feasible,scope] 欄序回答。", "en": "Declared nonempty interval record=[-22, 23], candidate=3/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. 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=[-22, 23]", "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