{"paper":{"title":"Generalized Dual Decomposition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Generalized dual decomposition replaces the linear regularizer with a nonlinear one to restore strong duality while keeping parallel subproblem solves in two-stage mixed-integer stochastic programs.","cross_cats":[],"primary_cat":"math.OC","authors_text":"Pengyu Zhang, Ruiwei Jiang","submitted_at":"2026-05-14T02:16:35Z","abstract_excerpt":"We study two-stage stochastic optimization models with mixed-integer decision variables appearing in both stages. For these models, dual decomposition enables parallel computing implementation and can quickly provide a lower bound for the optimal value. However, the lower bound thus obtained is not exact in general due to the lack of strong duality. In this paper, we propose a generalized dual decomposition (GDD) that extends the linear regularizer used in dual decomposition to a general nonlinear one, which still admits parallelization while exhibiting strong duality. By encoding the nonlinea"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we propose a generalized dual decomposition (GDD) that extends the linear regularizer used in dual decomposition to a general nonlinear one, which still admits parallelization while exhibiting strong duality... we establish the convergence of a GDD algorithm to achieve global optimum.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That nonlinear regularizers can be encoded through parameterization and cutting planes in a way that preserves both parallelizability and strong duality without introducing post-hoc data-dependent choices that invalidate the global-optimality guarantee.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Generalized Dual Decomposition replaces linear regularizers with nonlinear ones in dual decomposition to achieve strong duality and global optimality for mixed-integer two-stage stochastic programs while retaining parallelization.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Generalized dual decomposition replaces the linear regularizer with a nonlinear one to restore strong duality while keeping parallel subproblem solves in two-stage mixed-integer stochastic programs.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"5e05d054898f1bfd30ab3be7776a9295e04a6479c2db0e5b3f611948c2e684a2"},"source":{"id":"2605.14273","kind":"arxiv","version":1},"verdict":{"id":"e8c54fcf-7a78-490e-b25e-ca7223286593","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:49:21.756687Z","strongest_claim":"we propose a generalized dual decomposition (GDD) that extends the linear regularizer used in dual decomposition to a general nonlinear one, which still admits parallelization while exhibiting strong duality... we establish the convergence of a GDD algorithm to achieve global optimum.","one_line_summary":"Generalized Dual Decomposition replaces linear regularizers with nonlinear ones in dual decomposition to achieve strong duality and global optimality for mixed-integer two-stage stochastic programs while retaining parallelization.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That nonlinear regularizers can be encoded through parameterization and cutting planes in a way that preserves both parallelizability and strong duality without introducing post-hoc data-dependent choices that invalidate the global-optimality guarantee.","pith_extraction_headline":"Generalized dual decomposition replaces the linear regularizer with a nonlinear one to restore strong duality while keeping parallel subproblem solves in two-stage mixed-integer stochastic programs."},"references":{"count":17,"sample":[{"doi":"","year":null,"title":"Defineg ′ i(x) := PN i′=1 fi′(x) N −f i(x),∀i∈[N]","work_id":"906402d9-85f9-4e05-a41a-a0d709a5d1d8","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"inf x∈X lim infh(x) = inf x∈X limϵ→0 inf x′∈X∩B(x,ϵ) h(x′) ≥inf x∈X limϵ→0 (inf x′∈X h(x′)) = inf x∈X h(x)","work_id":"e09e17de-8f94-4dcd-ad95-395a00899b34","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Thus inf x∈X h(x) = infx∈X lim infh(x)","work_id":"1d631f4d-2e90-4422-acc5-d0f58a5ca7b7","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Thus,∥c(x i (r))− ˜xi∗∥ ≤ ∥c(xi (r))−x i (r)∥+∥x i (r) − ˜xi∗∥ ≤ ∥xi (R′) −x i (r)∥+∥x i (r) − ˜xi∗∥ ≤ 2ϵ 3 + ϵ 3 =ϵ. It follows that lim inf r→∞ gi(X t(r),x i (r)) = lim inf r→∞ g∗ i (c(xi (r))) (4) ","work_id":"ae7fdeb9-9a8b-4b86-a6ba-a491da8bd9d6","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Zhang and Jiang:Generalized Dual Decomposition 39","work_id":"13dee3c1-82ff-4475-a687-989a946a2125","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":17,"snapshot_sha256":"d5fe11ab4756564efba4239855135204387af85964107206f2b0988049520adf","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"8638753b0ee1312f44314799dc11d1993d330b00b080e3304ae9e6f7ef068c03"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}