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  "schema_version": 2,
  "title": "Reproduction: Provably Data-driven Lagrangian Relaxation for Mixed Integer Linear Programming",
  "emoji": "🎯",
  "space_id": "SabaPivot/repro-provably-data-driven-lagrangian-relaxation-for-mixed-integer-linear-programming",
  "paper": {
    "arxiv_id": "2605.19052"
  },
  "tags": [
    "icml2026-repro",
    "paper-OwLuqetJuB"
  ],
  "updated_at": "2026-07-27T14:29:12+00:00",
  "root": {
    "slug": "index",
    "title": "Reproduction: Provably Data-driven Lagrangian Relaxation for Mixed Integer Linear Programming",
    "file": "pages/index.md",
    "children": [
      {
        "slug": "executive-summary",
        "title": "Executive summary",
        "file": "pages/executive-summary/page.md",
        "children": []
      },
      {
        "slug": "claim-1-for-learned-lagrangian-relaxation-multipliers-over-s-coupling-constraints-and-n-training-samples-the-expected-excess-risk-of-the-erm-solution-is-upper-bounded-by-o-s-1-5-sqrt-n-theorem-5-5",
        "title": "Claim 1: For learned Lagrangian Relaxation multipliers over s coupling constraints and N training samples, the expected excess risk of the ERM solution is upper bounded by O(s^1.5/sqrt(N)) (Theorem 5.5).",
        "file": "pages/claim-1-for-learned-lagrangian-relaxation-multipliers-over-s-coupling-constraints-and-n-training-samples-the-expected-excess-risk-of-the-erm-solution-is-upper-bounded-by-o-s-1-5-sqrt-n-theorem-5-5/page.md",
        "children": []
      },
      {
        "slug": "claim-2-a-minimax-lower-bound-of-omega-s-sqrt-n-is-proven-showing-linear-dependence-on-the-number-of-coupled-constraints-s-is-unavoidable-for-any-learning-algorithm-theorem-5-6",
        "title": "Claim 2: A minimax lower bound of Omega(s/sqrt(N)) is proven, showing linear dependence on the number of coupled constraints s is unavoidable for any learning algorithm (Theorem 5.6).",
        "file": "pages/claim-2-a-minimax-lower-bound-of-omega-s-sqrt-n-is-proven-showing-linear-dependence-on-the-number-of-coupled-constraints-s-is-unavoidable-for-any-learning-algorithm-theorem-5-6/page.md",
        "children": []
      },
      {
        "slug": "claim-3-stochastic-gradient-ascent-with-iterate-averaging-achieves-the-matching-o-s-sqrt-n-rate-and-is-minimax-optimal-theorem-5-12",
        "title": "Claim 3: Stochastic Gradient Ascent with iterate averaging achieves the matching O(s/sqrt(N)) rate and is minimax optimal (Theorem 5.12).",
        "file": "pages/claim-3-stochastic-gradient-ascent-with-iterate-averaging-achieves-the-matching-o-s-sqrt-n-rate-and-is-minimax-optimal-theorem-5-12/page.md",
        "children": []
      },
      {
        "slug": "claim-4-learning-to-warm-start-subgradient-ascent-has-an-o-s-n-risk-upper-bound-and-matching-omega-s-n-minimax-lower-bound-theorems-6-1-and-6-2",
        "title": "Claim 4: Learning to warm-start subgradient ascent has an O(s/N) risk upper bound and matching Omega(s/N) minimax lower bound (Theorems 6.1 and 6.2).",
        "file": "pages/claim-4-learning-to-warm-start-subgradient-ascent-has-an-o-s-n-risk-upper-bound-and-matching-omega-s-n-minimax-lower-bound-theorems-6-1-and-6-2/page.md",
        "children": []
      },
      {
        "slug": "claim-5-the-multiplier-class-covering-number-and-its-entropy-integral-yield-rademacher-complexity-o-s-1-5-sqrt-n-lemmas-5-3-and-5-4",
        "title": "Claim 5: The multiplier-class covering number and its entropy integral yield Rademacher complexity O(s^1.5/sqrt(N)) (Lemmas 5.3 and 5.4).",
        "file": "pages/claim-5-the-multiplier-class-covering-number-and-its-entropy-integral-yield-rademacher-complexity-o-s-1-5-sqrt-n-lemmas-5-3-and-5-4/page.md",
        "children": []
      },
      {
        "slug": "claim-6-the-dual-function-is-concave-in-the-multiplier-and-has-subgradient-norm-at-most-2b-sqrt-s-under-bounded-violation-assumptions-proposition-5-1",
        "title": "Claim 6: The dual function is concave in the multiplier and has subgradient norm at most 2B sqrt(s) under bounded violation assumptions (Proposition 5.1).",
        "file": "pages/claim-6-the-dual-function-is-concave-in-the-multiplier-and-has-subgradient-norm-at-most-2b-sqrt-s-under-bounded-violation-assumptions-proposition-5-1/page.md",
        "children": []
      },
      {
        "slug": "conclusion",
        "title": "Conclusion",
        "file": "pages/conclusion/page.md",
        "children": []
      }
    ]
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