- The paper presents Mix-CALADIN, a novel algorithm that achieves consensus on mixed-integer problems without invoking local MIP solvers.
- It employs a two-stage approach: a continuous relaxation for initialization followed by a penalty-based refinement to enforce Boolean feasibility.
- Numerical results show linear convergence and improved performance over heuristic ADMM methods, highlighting its scalability and robustness.
Mix-CALADIN: Consensus Optimization for Mixed-Integer Programs Without Local MIP Solvers
Problem Overview and Motivation
The paper presents Mix-CALADIN, a distributed algorithm targeting consensus optimization problems with mixed-integer variables—particularly Boolean variables. Addressing a significant gap, Mix-CALADIN eschews the need for local mixed-integer solvers, which are typically invoked by agents in contemporary distributed mixed-integer programming (MIP) techniques. This design choice directly addresses the inherent computational and scalability limitations of centralized and agent-based MIP solvers, particularly as problem size and network scale induce prohibitive memory and runtime requirements.
From a distributed optimization perspective, the paper targets problems where each agent possesses a local objective depending on both continuous and Boolean variables and strives for consensus on a global decision. Prior methods either fall into centralized, heuristic, or decomposition approaches, with only limited distributed dual-decomposition results providing convergence guarantees—and those typically rely on local or centralized MIP subproblem solvers. Existing consensus-ADMM and ALADIN extensions successfully address continuous variables, but their extension to discrete settings has been nontrivial, with heuristic rounding or Big-M relaxations often sacrificing convergence guarantees and numerical robustness.
Mix-CALADIN Algorithm
Two-Stage Architecture
Mix-CALADIN builds on the Consensus Augmented Lagrangian Alternating Direction Inexact Newton (CALADIN) framework and is partitioned into a two-stage process:
Stage I - Relaxation and Initialization:
The mixed-integer consensus problem is first relaxed by treating all Boolean variables as continuous. The consensus-ALADIN (CALADIN) method then solves this continuous relaxation, providing a theoretical lower bound to the original problem and a high-quality initialization for the subsequent stage. The agents independently solve their local (now continuous) subproblems, and the coordinator aggregates gradients and Hessians to update the global variables.
Stage II - Integer Feasibility and Penalty Enforcement:
Upon convergence of the relaxed problem, the solution is refined towards Boolean feasibility. Rather than introducing a nonconvex penalty, at each iteration the coordinator solves a box-constrained QP with a stagewise-increasing linear penalty term that, in expectation, steers relaxed variables to Boolean values. The penalty parameter is increased only when the Boolean constraint is insufficiently enforced, resulting in a stable bi-level iteration where the inner-loop solves a smooth quadratic program and the outer-loop escalates the penalty for infeasibility.
Absence of Local MIP Solvers
A key distinctive property and strong claim of the algorithm is that Mix-CALADIN never invokes local mixed-integer solvers at any point. Instead, only continuous NLP/QP subproblems and consensus quadratic programs are solved—this property holds irrespective of convexity or nonconvexity in local agent objectives.
Theoretical Guarantees
- Stage I:
- Convex Case: Global linear convergence is established for smooth, strongly convex objectives.
- Nonconvex Case: Local linear convergence is achieved under suitable regularity and local convexity (second-order sufficient conditions).
- Stage II:
- Under mild Lipschitz continuity assumptions and penalty parameters exceeding the agent gradient Lipschitz constant, the augmented energy function is shown to decrease monotonically, guaranteeing convergence to Boolean consensus solutions.
These results are supported with formal theorems, leveraging sequential quadratic surrogates and penalty argumentation for discrete feasibility.
Numerical Results
The paper presents comprehensive numerical studies on two distributed consensus problems:
- Nonconvex sensor localization with mixed Boolean/continuous variables: Mix-CALADIN demonstrates robust convergence for 20 agents, each with 10 continuous and 10 Boolean variables, and efficiently handles the nonconvex coupling in objectives.
- Benchmarking with convex consensus problems: Comparison with a state-of-the-art projection-based ADMM (Takapoui et al., 2020) shows that Mix-CALADIN consistently achieves better objective values and exact satisfaction of Boolean constraints, whereas the ADMM-based approach is heuristic and lacks convergence guarantees.
Notably, theoretical convergence rates are corroborated by the observed linear decay in primal residuals, and the two-stage procedure yields solutions that strictly satisfy the discrete constraints—contradicting the necessity for local MIP solvers in existing literature.
Practical and Theoretical Implications
Algorithmic Efficiency and Applicability
Mix-CALADIN enables scalable solution of large-scale distributed MIPs by eliminating the bottleneck of agent-side MIP solvers. This property is highly impactful for real-time applications (e.g., transport network design, sensor scheduling, combinatorial resource allocation in power grids) where solver overheads and communication costs are critical and Boolean constraints are prevalent.
The staged relaxation and penalty strategy provides both deterministic quality bounds (via relaxation) and discrete feasibility—without resorting to problem-specific heuristics or Big-M methods that often destabilize distributed optimization.
Theoretical Perspective
The work closes a key gap in distributed optimization. Prior approaches either invoked local MIP solvers or lacked convergence guarantees when handling discrete variables in consensus algorithms. Mix-CALADIN demonstrates that, by leveraging sequential quadratic penalty update schemes and continuous relaxations, distributed consensus on Boolean variables is achievable with formal convergence rates and no reliance on discrete subproblem solvers.
Future Directions
The clear modularity of Mix-CALADIN makes it amenable to further acceleration (e.g., second-order method in Stage II, adaptive penalty update schemes), deployment in partially synchronous or lossy communication networks, and extension to more general discrete variable domains beyond Boolean. The algorithmic template could inspire developments for federated learning with discrete constraints, distributed Nash equilibrium seeking over mixed-variable domains, and robust multi-agent coordination.
Conclusion
Mix-CALADIN introduces a scalable, solver-agnostic, and theoretically-grounded approach for consensus mixed-integer optimization in distributed multi-agent systems (2604.14897). The algorithm convincingly demonstrates that decentralized solution of nonconvex, combinatorial consensus problems with Boolean variables is feasible without invoking local MIP solvers, and achieves strong convergence guarantees and solution quality superior to prevalent heuristic alternatives. This work redefines tractable architectures for distributed MIP and establishes a new baseline for theory and practice in this domain.