We propose a general algorithm of constructing an extended formulation for any given set of linear constraints with integer coefficients. Our algorithm consists of two phases: first construct a decision diagram $(V,E)$ that somehow represents a given $m \times n$ constraint matrix, and then build an equivalent set of $|E|$ linear constraints over $n+|V|$ variables. That is, the size of the resultant extended formulation depends not explicitly on the number $m$ of the original constraints, but on its decision diagram representation. Therefore, we may significantly reduce the computation time for optimization problems with integer constraint matrices by solving them under the extended formulations, especially when we obtain concise decision diagram representations for the matrices. We can apply our method to $1$-norm regularized hard margin optimization over the binary instance space $\{0,1\}^n$, which can be formulated as a linear programming problem with $m$ constraints with $\{-1,0,1\}$-valued coefficients over $n$ variables, where $m$ is the size of the given sample. Furthermore, introducing slack variables over the edges of the decision diagram, we establish a variant formulation of soft margin optimization. We demonstrate the effectiveness of our extended formulations for integer programming and the $1$-norm regularized soft margin optimization tasks over synthetic and real datasets.

翻译：我们提出一种通用算法，用于为任意给定整数系数线性约束集构造扩展表述。该算法包含两个阶段：首先构建一个决策图$(V,E)$以某种方式表示给定的$m \times n$约束矩阵，再建立一组等价的、包含$n+|V|$个变量的$|E|$个线性约束。即最终扩展表述的规模不直接依赖于原始约束数量$m$，而取决于其决策图表示。因此，通过将整数约束矩阵的优化问题转化为扩展表述进行求解，可显著降低计算时间——尤其当获得矩阵的紧致决策图表示时。我们的方法可应用于二元实例空间$\{0,1\}^n$上的$1$-范数正则化硬间隔优化问题，该问题可表述为包含$m$个约束（系数取自$\{-1,0,1\}$）和$n$个变量的线性规划，其中$m$为给定样本规模。进一步地，通过在决策图边上引入松弛变量，我们建立了软间隔优化的变体表述。基于合成数据集与真实数据集的实验表明，该扩展表述在整数规划及$1$-范数正则化软间隔优化任务中具有显著效果。