Facial reduction for symmetry reduced semidefinite and doubly nonnegative programs
研究了面约化与对称约化在半定规划中的结合，发现两者在交替方向乘子法中配合良好，能有效处理双非负松弛，显著提升数值稳定性和运行速度，并测试了n=500的二次分配问题。
Abstract 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 $$n=500$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>500</mml:mn> </mml:mrow> </mml:math> . This translates to a semidefinite constraint of order 250, 000 and $$625\times 10^8$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>625</mml:mn> <mml:mo>×</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>8</mml:mn> </mml:msup> </mml:mrow> </mml:math> nonnegative constrained variables, before applying the reduction techniques.