Integer linear programming (ILP) models a wide range of practical combinatorial optimization problems and has significant impacts in industry and management sectors. This work proposes new characterizations of ILP with the concept of boundary solutions. Motivated by the new characterizations, we develop an efficient local search solver, which is the first local search solver for general ILP validated on a large heterogeneous problem dataset. We propose a new local search framework that switches between three modes, namely Search, Improve, and Restore modes. We design tailored operators adapted to different modes, thus improving the quality of the current solution according to different situations. For the Search and Restore modes, we propose an operator named tight move, which adaptively modifies variables' values, trying to make some constraint tight. For the Improve mode, an efficient operator lift move is proposed to improve the quality of the objective function while maintaining feasibility. Putting these together, we develop a local search solver for integer linear programming called Local-ILP. Experiments conducted on the MIPLIB dataset show the effectiveness of our solver in solving large-scale hard integer linear programming problems within a reasonably short time. Local-ILP is competitive and complementary to the state-of-the-art commercial solver Gurobi and significantly outperforms the state-of-the-art non-commercial solver SCIP. Moreover, our solver establishes new records for 6 MIPLIB open instances. The theoretical analysis of our algorithm is also presented, which shows our algorithm could avoid visiting unnecessary regions and also maintain good connectivity of targeted solutions.

翻译：整数线性规划（ILP）建模了广泛的实用组合优化问题，并在工业和管理领域具有显著影响。本文利用边界解的概念提出了ILP的新刻画。受这些新刻画的启发，我们开发了一种高效的局部搜索求解器，这是首个在大型异构问题数据集上验证的通用ILP局部搜索求解器。我们提出了一种新的局部搜索框架，该框架在三种模式之间切换，即搜索模式、改进模式和恢复模式。我们设计了适应不同模式的定制化算子，从而根据具体情况提升当前解的质量。对于搜索模式和恢复模式，我们提出了名为"紧移动"的算子，该算子自适应地修改变量的值，试图使某些约束达到紧致状态。对于改进模式，我们提出了一种高效的"提升移动"算子，在保持可行性的同时提升目标函数的质量。综合以上方法，我们开发了一个名为Local-ILP的整数线性规划局部搜索求解器。在MIPLIB数据集上的实验表明，我们的求解器能在合理时间内有效求解大规模困难整数线性规划问题。Local-ILP与最先进的商业求解器Gurobi相比具有竞争力和互补性，并显著优于最先进的非商业求解器SCIP。此外，我们的求解器为6个MIPLIB开放实例创造了新纪录。本文还给出了算法的理论分析，表明我们的算法能够避免访问不必要的区域，并保持目标解的良好连通性。