Facial reduction for symmetry reduced semidefinite and doubly nonnegative programs.

We consider both facial reduction, FR, and symmetry reduction, SR, techniques for semidefinite programming, SDP. We show that the two together fit surprisingly well in an alternating direction method of multipliers, ADMM, approach. In fact, this approach allows for simply adding on nonnegativity constraints, and solving the doubly nonnegative, DNN , relaxation of many classes of hard combinatorial problems. We also show that the singularity degree remains the same after SR, and that the DNN relaxations considered here have singularity degree one, that is reduced to zero after FR. The combination of FR and SR leads to a significant improvement in both numerical stability and running time for both the ADMM and interior point approaches. We test our method on various DNN relaxations of hard combinatorial problems including quadratic assignment problems with sizes of more than . This translates to a semidefinite constraint of order 250, 000 and nonnegative constrained variables, before applying the reduction techniques.

我们考虑半定规划（SDP）的面约简（FR）和对称约简（SR）技术。我们表明，这两种技术在交替方向乘子法（ADMM）中结合得非常好。实际上，这种方法允许简单地添加非负性约束，并求解许多类困难组合问题的双非负（DNN）松弛。我们还表明，在SR之后奇异度保持不变，并且这里考虑的DNN松弛具有奇异度1，在FR之后降为0。FR和SR的结合使得ADMM和内点法在数值稳定性和运行时间上都有显著提高。我们在各种困难组合问题的DNN松弛上测试我们的方法，包括规模大于[此处可能缺失具体规模数值]的二次分配问题。在应用约简技术之前，这对应于一个250,000阶的半定约束和[此处可能缺失变量数量相关内容]个非负约束变量。