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Survey on Lagrangian relaxation for MILP: importance, challenges, historical review, recent advancements, and opportunities

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Abstract

Operations in areas of importance to society are frequently modeled as mixed-integer linear programming (MILP) problems. While MILP problems suffer from combinatorial complexity, Lagrangian Relaxation has been a beacon of hope to resolve the associated difficulties through decomposition. Due to the non-smooth nature of Lagrangian dual functions, the coordination aspect of the method has posed serious challenges. This paper presents several significant historical milestones (beginning with Polyak’s pioneering work in 1967) toward improving Lagrangian Relaxation coordination through improved optimization of non-smooth functionals. Finally, this paper presents the most recent developments in Lagrangian Relaxation for fast resolution of MILP problems. The paper also briefly discusses the opportunities that Lagrangian Relaxation can provide at this point in time.

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Notes

  1. In this particular case, \(f(x^k,y^k) \equiv \sum _{i=1}^I \left( (c_i^{x})^T \cdot x_i^k +(c_i^{y})^T \cdot y_i^k \right) \) such that \(\{x^k,y^k\}\) satisfy constraints (2).

  2. Dual values can be used to quantify the quality of the solution \(\{x^k,y^k\}\).

  3. To solve MILP problems, Lagrangian Relaxation is often regarded as a heuristic. However, in dual space, the Lagrangian relaxation method is exact; the method is also capable of helping to improve solutions through the multipliers update, unlike many other heuristic methods.

  4. Superlinear convergence is also possible, however, (1) a reformulation of the dual problem (Charisopoulos and Davis, 2022) is required; (2) within the Lagrangian Relaxation framework, a dual function is generally unavailable as argued in Sect. 2.4.

  5. Upon visual examination of the dual function illustrated in Fig. 1, the condition number is likely 0 since the subgradient emanating from point B appears to form a right angle with the direction toward optimal multipliers.

  6. A quality measure to quantify the quality of multipliers (i.e., how close the multipliers are to their optimal values) will be discussed in Sect. 2.9

  7. The parameter \(\rho \) is increased as \(\rho ^{k+1}=\beta \cdot \rho ^k,\beta >1\) until surrogate optimality condition (21) is no longer satisfied, which signifies that proper “surrogate” subgradient directions can no longer be obtained. Moreover, with heavily penalized constraint violations, the feasibility is overemphasized and subproblem solutions may get stuck at a suboptimal solution. In these situations, \(\rho \) is no longer increased, rather, violations of surrogate optimality conditions prompt the reduction of penalty coefficients \(\rho ^{k+1}=\rho ^k/\beta ,\beta >1\).

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Acknowledgements

This work was supported in part by the U.S. National Science Foundation under Grant ECCS-1810108.

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  1. Department of Electrical and Computer Engineering, University of Connecticut, Storrs, USA

    Mikhail A. Bragin

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Bragin, M.A. Survey on Lagrangian relaxation for MILP: importance, challenges, historical review, recent advancements, and opportunities. Ann Oper Res 333, 29–45 (2024). https://doi.org/10.1007/s10479-023-05499-9

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