二元函数上图的稀疏多项析取割
Sparse multi-term disjunctive cuts for the epigraph of a function of binary variables
摘要：
提出了一种新的二元函数上图有效不等式的分离方法。所提出的不等式是通过枚举二元变量的子集$I$获得的析取项定义的析取割。我们表明，通过限制支持的削减到相同的一组变量$I$，削减可以通过求解线性规划与$2^{|I|}$约束。虽然这限制了用于定义多项析取的集合$I$的大小，但是该过程使得能够使用比现有方法多得多的项来生成多项析取切割。我们提出了两种方法来选择变量的子集。使用$|I|$直到大小10的块对角结构的三个MILP问题的经验表明，稀疏切割通常可以关闭几乎与多项析取切割一样多的间隙，而没有这种限制，并且在一小部分时间内。我们还发现，包括这些削减这些MILP问题的切割和分支解决方案的方法，导致显着减少解决方案的时间或结束最优差距的情况下，没有解决的时间限制。最后，我们描述了如何所提出的方法可以适应最佳的“倾斜”一个给定的有效的不等式，通过修改系数的稀疏子集的变量。
英文摘要：
We propose a new method for separating valid inequalities for the epigraph of a function of binary variables. The proposed inequalities are disjunctive cuts defined by disjunctive terms obtained by enumerating a subset $I$ of the binary variables. We show that by restricting the support of the cut to the same set of variables $I$, a cut can be obtained by solving a linear program with $2^{|I|}$ constraints. While this limits the size of the set $I$ used to define the multi-term disjunction, the procedure enables generation of multi-term disjunctive cuts using far more terms than existing approaches. We present two approaches for choosing the subset of variables. Experience on three MILP problems with block diagonal structure using $|I|$ up to size 10 indicates the sparse cuts can often close nearly as much gap as the multi-term disjunctive cuts without this restriction and in a fraction of the time. We also find that including these cuts within a cut-and-branch solution method for these MILP problems leads to significant reductions in solution time or ending optimality gap for instances that were not solved within the time limit. Finally, we describe how the proposed approach can be adapted to optimally "tilt" a given valid inequality by modifying the coefficients of a sparse subset of the variables.
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