Many important optimization problems, such as manufacturing scheduling and power system unit commitment, are formulated as Mixed-Integer Linear Programming (MILP) problems. Such problems are generally difficult to solve because of their combinatorial nature, and may subject to strict computation time limitations. Recently, our decomposition-and-coordination method “Surrogate Absolute Value Lagrangian Relaxation” (SAVLR) exploits the exponential reduction of complexity upon problem decomposition and effectively coordinates subproblem solutions. In the method, subproblems are generally solved by using Branch-and-Cut (B&C). When subproblems are complicated, however, the approach might not be able to generate high-quality solutions within time limitations. In this paper, motivated by the “Ordinal Optimization” concept, this difficulty is resolved through exploiting a specific property of SAVLR that subproblem solutions only need to be “good enough” to satisfy a convergence condition. Time consuming B&C is eliminated in many iterations through obtaining “good enough” subproblem solutions based on “crude models” (e.g., LP-relaxed problems) or from heuristics. Testing results on generalized assignment problems demonstrate that the approach obtains high-quality solutions in a computationally efficient manner and significantly outperforms other approaches. This approach also opens up a new way to solve practical MILP problems that are subject to strict computation time limitations.

中文翻译：

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序数优化概念使混合整数线性规划问题的分解和协调成为可能

许多重要的优化问题，例如制造调度和电力系统单元承诺，都被表述为混合整数线性规划 (MILP) 问题。由于它们的组合性质，此类问题通常难以解决，并且可能受到严格的计算时间限制。最近，我们的分解和协调方法“代理绝对值拉格朗日松弛”（SAVLR）利用问题分解时复杂性的指数降低并有效地协调子问题的解决方案。在该方法中，子问题一般采用Branch-and-Cut (B&C) 来解决。然而，当子问题很复杂时，该方法可能无法在时间限制内生成高质量的解决方案。在本文中，受“序数优化”概念的启发，这个困难是通过利用 SAVLR 的一个特定属性来解决的，即子问题的解决方案只需要“足够好”来满足收敛条件。通过基于“原始模型”（例如，LP 松弛问题）或启发式获得“足够好”的子问题解决方案，在许多迭代中消除了耗时的 B&C。广义分配问题的测试结果表明，该方法以计算效率高的方式获得了高质量的解决方案，并且明显优于其他方法。这种方法还开辟了一种新方法来解决受到严格计算时间限制的实际 MILP 问题。通过基于“原始模型”（例如，LP 松弛问题）或启发式获得“足够好”的子问题解决方案，C 在许多迭代中被消除。广义分配问题的测试结果表明，该方法以计算效率高的方式获得了高质量的解决方案，并且明显优于其他方法。这种方法还开辟了一种新方法来解决受到严格计算时间限制的实际 MILP 问题。通过基于“粗略模型”（例如，LP 松弛问题）或启发式获得“足够好”的子问题解决方案，在许多迭代中消除了 C。广义分配问题的测试结果表明，该方法以计算效率高的方式获得了高质量的解决方案，并且明显优于其他方法。这种方法还开辟了一种新方法来解决受到严格计算时间限制的实际 MILP 问题。