{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "2JS8PPP8XW"
    },
    {
      "field": "uses",
      "target": "ZBVJ4B41AX"
    },
    {
      "field": "defines",
      "target": "32CP806EHX"
    }
  ],
  "body": "\n## Summary\n\nLinear programming is the optimization of a linear function over the set of solutions of finitely many linear inequalities.\nThe feasible set is a convex polyhedron.\nWhen an optimum exists, it is attained on a face, and at a vertex when the polyhedron has vertices.\nEach linear program has a dual program.\nIf one of the two has an optimal solution, so does the other, with the same optimal value; this is tied to [[Farkas' lemma|FYZC634H2D]].\nLinear programs are solved in practice by the [[simplex algorithm|MH272QHWPM]] and by interior point methods.\nThe ellipsoid method and interior point methods solve them in polynomial time.\n\n## Explanations\n\n### L1 human\n\nA factory makes two products, each using some amounts of limited resources, and earns some profit on each.\nHow much of each should it make to earn the most?\nThe limits are straight-line conditions, so the allowed plans form a polygon, and the best plan is found at one of its corners.\nLinear programming is the study of such problems, with any number of variables and conditions.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "90C05",
      "52B55"
    ],
    "name": "Linear programming",
    "same_as": {
      "wikidata": "Q202843",
      "wikipedia": "Linear programming"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Linear_programming"
      }
    ],
    "tag": "VNMXB0J2KA"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/VNMXB0J2KA/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/VNMXB0J2KA.md",
  "msc": [
    "90C05",
    "52B55"
  ],
  "name": "Linear programming",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VNM/VNMXB0J2KA.md",
  "status": null,
  "tag": "VNMXB0J2KA",
  "uri": "m2://VNMXB0J2KA",
  "url": "https://tamnd.github.io/math2/wiki/linear-programming/",
  "writing": {
    "tag": "VNMXB0J2KA",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}