{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-optimization-problem",
  "semantic_family_id": "sf-optimization-problem",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "最佳化問題"
  },
  "summary": {
    "zh-Hant": "最佳化問題在明示可行集合上尋找使目標函數最小或最大的點；最優值與最優解集合需分開。"
  },
  "statement": {
    "zh-Hant": "給可行集合 S 與目標函數 f:S→R，最小化問題尋找 x*∈S，使每個 x∈S 都有 f(x*)≤f(x)；最大化反向。最優值可能由多個最優點同時達成。"
  },
  "formula": {
    "tex": "f(s)=m",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>f</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>=</mo><mi>m</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "f",
          "glyph": "f"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "s",
            "glyph": "s"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "m",
        "glyph": "m"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "29537024f78f86b81b7a685d0394740ea7148aa6648a59f2a39134d0c39daff1"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫把有限候選、可行遮罩、限制與目標值完整列出。"
    },
    {
      "id": "a2",
      "zh-Hant": "枚舉所得結論只涵蓋宣告的有限可行集合，不外推未列候選或一般計算複雜度。"
    }
  ],
  "symbols": [
    {
      "token": "f",
      "role": "objective_function",
      "role_zh": "目標函數",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "feasible_candidate",
      "role_zh": "可行候選",
      "scope": "representative_formula"
    },
    {
      "token": "m",
      "role": "optimal_value",
      "role_zh": "最優值",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-set",
      "title_zh": "集合",
      "reason_zh": "可行域是候選點集合。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "目標函數為每個可行點指定數值。"
    },
    {
      "id": "mko-order-relation",
      "title_zh": "序關係",
      "reason_zh": "最小值與最大值依序比較。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "先判定可行性，再比較目標值；不可行點即使數值更好也不能成為最優解。",
      "最優值是一個數，argmin 或 argmax 是所有達成該值的可行點集合，可能有平手。",
      "完整枚舉可以證明有限清單內的最優性，但不說明連續可行域、無界問題或大型問題的求解效率。"
    ],
    "common_misconception_zh": "忽略限制直接挑最小數，或只回報一個平手解並宣稱唯一。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪只證完整明示有限候選集內的最優性；不宣稱一般線性或非線性規劃已求解。",
    "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": "建立線性規劃、可行多面體、對偶與最優性證書。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-optimization-problem.md 整理；2026-10-13 脈衝 R04（使用者手動接續 2026-10-20 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"
  }
}