{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-linear-programming",
  "semantic_family_id": "sf-linear-programming",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "線性規劃"
  },
  "summary": {
    "zh-Hant": "線性規劃在線性等式或不等式限制的可行域上最佳化線性目標函數；本輪題庫只驗完整有限候選網格。"
  },
  "statement": {
    "zh-Hant": "線性規劃由線性目標與線性限制構成，尋找可行域上的最小或最大值。有限候選網格可以完整核對列出的點，但除非已證候選完整涵蓋一般可行域，結論只能稱為所給候選中的最佳。"
  },
  "formula": {
    "tex": "L(x,y)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>L</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "L",
        "glyph": "L"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "x",
          "glyph": "x"
        },
        {
          "type": "symbol",
          "name": "y",
          "glyph": "y"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "91ece290372873ddcad439f03283d7d6e2c00bd024c12b0de112b8d7f5a70b73"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫使用二變數整係數限制 ax+by≤c、整係數目標及完整明示有限候選點。"
    },
    {
      "id": "a2",
      "zh-Hant": "不以有限格點枚舉宣稱連續多面體或一般LP的全域最優解。"
    }
  ],
  "symbols": [
    {
      "token": "L",
      "role": "linear_program",
      "role_zh": "線性規劃資料",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "first_coordinate",
      "role_zh": "第一決策座標",
      "scope": "representative_formula"
    },
    {
      "token": "y",
      "role": "second_coordinate",
      "role_zh": "第二決策座標",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-optimization-problem",
      "title_zh": "最佳化問題",
      "reason_zh": "LP是線性目標與線性限制的最佳化特例。"
    },
    {
      "id": "mko-linear-equation",
      "title_zh": "一次方程式",
      "reason_zh": "限制邊界使用線性式。"
    },
    {
      "id": "mko-system-linear-equations",
      "title_zh": "二元一次聯立方程組",
      "reason_zh": "限制邊界交點可由線性系統描述。"
    },
    {
      "id": "mko-vector",
      "title_zh": "向量",
      "reason_zh": "目標與限制係數可視為向量內積。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "先逐限制核對可行性，再只在可行候選間比較目標值。",
      "線性目標可能在多個候選點平手，argmin或argmax應保留全部索引。",
      "有限候選表的完整性是結論範圍的一部分；沒有多面體頂點完備性證明時不能外推。"
    ],
    "common_misconception_zh": "把不可行但目標值較好的點當最優，或把有限格點最佳冒充一般LP最優。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只證所給有限候選中的最佳；一般LP可行域、對偶與最優證書未建立。",
    "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": "加入標準形、頂點、對偶與可重播最優性證書。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-linear-programming.md 整理；2026-10-13 脈衝 R06（使用者手動接續 2026-10-21 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"
  }
}