Mix-CALADIN: A Distributed Algorithm for Consensus Mixed-Integer Optimization

Apostolos I. Rikos; Boyu Han; Karl H. Johansson; Xu Du

arxiv: 2604.14897 · v1 · submitted 2026-04-16 · 🧮 math.OC · cs.SY· eess.SY

Mix-CALADIN: A Distributed Algorithm for Consensus Mixed-Integer Optimization

Boyu Han , Xu Du , Karl H. Johansson , Apostolos I. Rikos This is my paper

Pith reviewed 2026-05-10 10:56 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY

keywords distributed optimizationmixed-integer programmingBoolean variablesconsensus algorithmsaugmented Lagrangianconvergence analysisalternating direction method

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The pith

Mix-CALADIN extends CALADIN to solve distributed consensus problems with Boolean variables and supplies convergence guarantees for convex and nonconvex cases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents Mix-CALADIN, a distributed algorithm for consensus optimization problems that include mixed-integer variables, with a focus on Boolean variables. It modifies the existing CALADIN framework by adding specialized handling techniques that avoid any need for local mixed-integer solvers at each node. Rigorous convergence is shown for both convex and nonconvex problems whenever the objective functions meet a Lipschitz continuity condition. This matters for large-scale systems where agents must agree on decisions that include discrete choices, because the method supplies theoretical backing without requiring centralized computation or per-agent integer programming routines. Experiments indicate the approach matches or exceeds prior methods in speed while adding the missing guarantees.

Core claim

Mix-CALADIN extends the Consensus Augmented Lagrangian Alternating Direction Inexact Newton (CALADIN) framework by incorporating specialized techniques for handling Boolean variables without relying on local mixed-integer solvers, and proves convergence for both convex and nonconvex mixed-integer consensus optimization problems under the assumption that the objective functions are Lipschitz continuous.

What carries the argument

The Mix-CALADIN algorithm, which augments the CALADIN framework with Boolean-variable handling rules inside an augmented Lagrangian alternating-direction iteration.

Load-bearing premise

The objective functions satisfy Lipschitz continuity.

What would settle it

A concrete counterexample consisting of a distributed consensus problem with Lipschitz-continuous objectives where the Mix-CALADIN iterates fail to converge would falsify the claimed guarantees.

Figures

Figures reproduced from arXiv: 2604.14897 by Apostolos I. Rikos, Boyu Han, Karl H. Johansson, Xu Du.

**Figure 4.** Figure 4: Evolution of the energy function E(z [k] ) during Stage II of Algorithm 1 for the nonconvex problem (17). exact Boolean constraint in Stage II leads to a temporary increase in the objective value, reflecting the formulation’s shift towards feasibility. After 199 iterations, Algorithm 1 converges to a solution that is superior to that attained by the projection-based ADMM approach, demonstrating both improv… view at source ↗

**Figure 5.** Figure 5: Convergence comparison of Algorithm 1 and [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

read the original abstract

This paper addresses distributed consensus optimization problems with mixed-integer variables, with a specific focus on Boolean variables. We introduce a novel distributed algorithm that extends the Consensus Augmented Lagrangian Alternating Direction Inexact Newton (CALADIN) framework by incorporating specialized techniques for handling Boolean variables without relying on local mixed-integer solvers. Under the mild assumption of Lipschitz continuity of the objective functions, we establish rigorous convergence guarantees for both convex and nonconvex mixed-integer programming problems. Numerical experiments demonstrate that the proposed algorithm achieves competitive performance compared to existing approaches while providing rigorous convergence guarantees.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Mix-CALADIN extends CALADIN to handle Boolean variables in distributed consensus optimization without local solvers and claims convergence for convex and nonconvex cases under Lipschitz continuity.

read the letter

Mix-CALADIN takes the existing CALADIN approach and adds handling for Boolean variables in distributed consensus optimization. It avoids using local mixed-integer solvers at each node, which is a practical move for networked systems. The new part is the specialized techniques for Booleans and the convergence results that cover both convex and nonconvex problems. The paper shows competitive numerical performance against other methods. That combination of guarantees and no local solvers is the useful bit for applications in control and allocation. The convergence rests on Lipschitz continuity of the objectives, which is a standard condition but carries a lot of the proof. The abstract mentions rigorous guarantees, but without seeing the full update rules and derivations it's tough to verify how the inexact steps and discrete handling fit together. The experiments are described as competitive, yet more detail on the test cases would help assess how general the results are. This work fits for researchers in distributed optimization who deal with integer constraints in multi-agent setups. Someone looking for an implementable algorithm with some theory backing would get value from it, though it builds directly on prior CALADIN work rather than starting from scratch. I would send this to peer review. The claims are concrete and the area is active enough that referees can check the proofs and experiments properly.

Referee Report

0 major / 3 minor

Summary. The paper introduces Mix-CALADIN, a distributed algorithm extending the CALADIN framework to solve consensus optimization problems with mixed-integer (specifically Boolean) variables. It handles Boolean variables via specialized techniques that avoid local mixed-integer solvers. Under the assumption of Lipschitz continuity of the objective functions, the manuscript claims rigorous convergence guarantees for both convex and nonconvex mixed-integer programming problems, supported by numerical experiments showing competitive performance relative to existing methods.

Significance. If the stated convergence results hold, the work provides a valuable contribution to distributed optimization by delivering a scalable algorithm with theoretical guarantees for mixed-integer consensus problems, an area where such guarantees are difficult to obtain. The extension of CALADIN without reliance on local solvers and the coverage of nonconvex cases represent clear strengths. The explicit invocation of a standard Lipschitz assumption to control inexact steps is appropriately highlighted.

minor comments (3)

[Algorithm description] The description of the Boolean-variable update rules in the algorithm section would benefit from an explicit pseudocode listing or step-by-step enumeration to improve readability and reproducibility.
[Numerical experiments] Numerical experiments section: the problem dimensions, number of agents, and specific instance generation details are not fully specified, which hinders direct reproduction of the reported competitive performance.
[Figures] Convergence plots in the experiments would be clearer if they included results from multiple random seeds or error bands rather than single-run trajectories.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The referee summary accurately captures the key elements of Mix-CALADIN, including the extension of the CALADIN framework, handling of Boolean variables without local solvers, and convergence guarantees under Lipschitz continuity for both convex and nonconvex cases.

Circularity Check

0 steps flagged

No significant circularity; convergence tied to external Lipschitz assumption

full rationale

The paper's central result is a convergence guarantee for the Mix-CALADIN algorithm under the explicit external assumption of Lipschitz continuity of the objective functions. This assumption is standard in the literature for ADMM-style methods and is not derived from or fitted to the algorithm's outputs. The extension for Boolean variables is described without reducing any key step to self-citation chains, self-definitional loops, or renaming of known results. The derivation remains self-contained against external benchmarks, with no load-bearing internal reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on extending an existing framework and on one standard domain assumption; no free parameters or invented entities are mentioned.

axioms (1)

domain assumption Lipschitz continuity of the objective functions
Invoked explicitly to establish convergence for convex and nonconvex cases.

pith-pipeline@v0.9.0 · 5396 in / 1192 out tokens · 34101 ms · 2026-05-10T10:56:49.338036+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Distributed and Decentralized Optimization Algorithms via Consensus ALADIN
math.OC 2026-05 unverdicted novelty 6.0

The paper proposes Consensus ALADIN (C-ALADIN) algorithms that solve distributed consensus optimization with global convergence for convex problems and local convergence for non-convex ones, including a decentralized ...

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages · cited by 1 Pith paper

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