{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-convex-optimization-problem",
  "semantic_family_id": "sf-convex-optimization-problem",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "凸最佳化問題"
  },
  "summary": {
    "zh-Hant": "凸最佳化問題在凸可行域上最小化凸目標；問題資格、可行性、求值與全域最優是不同責任。"
  },
  "statement": {
    "zh-Hant": "最小化問題只有在目標函數凸且可行域凸時才具本條目的凸問題資格。本輪只核一元區間與二次目標的資格、單點可行性、單點求值及兩個可行候選的局部比較。"
  },
  "formula": {
    "tex": "C(P)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>C</mi><mo>(</mo><mi>P</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "C",
        "glyph": "C"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "P",
          "glyph": "P"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "3bb09c67eb0548e66d16a1aabab6ff91f93698d62d6aeeb7fe127c717848b308"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "可行域為明示實區間，目標為整係數二次函數。"
    },
    {
      "id": "a2",
      "zh-Hant": "不由資格、單點或兩點比較宣稱已找到全域解；不引入KKT、通用求解器或收斂結果。"
    }
  ],
  "symbols": [
    {
      "token": "C",
      "role": "convex_problem_record",
      "role_zh": "凸問題資格紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "objective_function",
      "role_zh": "目標函數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-convex-function",
      "title_zh": "凸函數",
      "reason_zh": "目標凸性是凸最佳化資格之一。"
    },
    {
      "id": "mko-optimization-problem",
      "title_zh": "最佳化問題",
      "reason_zh": "可行域、目標與最小化語義來自一般最佳化。"
    },
    {
      "id": "mko-real-vector-space",
      "title_zh": "實向量空間",
      "reason_zh": "可行域位於實向量空間。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "目標凸與可行域凸必須分別核對。",
      "可行點判定與目標函數求值不等於最優性。",
      "兩候選比較只說兩者之間誰較低，不外推全域最優。"
    ],
    "common_misconception_zh": "看到凸字樣就宣稱有唯一全域解，或把兩候選中較低者叫整個問題的最優解。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理問題資格、可行性、求值與兩點比較；不宣稱求解。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入最優性證書、KKT、對偶與演算法收斂。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-convex-optimization-problem.md 整理；2026-10-13 脈衝 R03（使用者手動接續 2026-10-25 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}