论文标题
对称锥的共阳性锥的近似层次结构及其比较
Approximation hierarchies for copositive cone over symmetric cone and their comparison
论文作者
论文摘要
我们首先提供了由一般对称锥上共阳性（COP）锥体的约束（SOS）约束所描述的内部附属层次结构。层次结构是Parrilo（2000）对通常的COP锥（非负矫正）提出的概括。我们还讨论了它的双重。其次，我们使用通常的COP锥在对称锥上表征COP锥。通过更换De Klerk and Pasechnik（2002）或Yildirim（2012）提供的内部或外部或外部及时的层次结构，我们通过半内或外ximimation层次结构获得了半决赛，但不是由半际构造的，但不是由SOS构造的cop Matrix and Cone and and and cone cone and cone and cone cone and cone and cone cone and cone a cone and cone and Approxiimation层次结构，我们就获得了cone and and conter and and cont。然后，我们将它们与Zuluaga等人提供的现有层次结构进行了比较。 （2006年）和Lasserre（2014）。理论和数值检查意味着我们可以在数值上增加一个深度参数，该参数确定了近似层次结构在de Klerk和Pasechnik（2002）和Yildirim（2012）中的近似层次结构中，尤其是当非负矫正器很小时。在这种情况下，从Yildirim（2012）得出的近似层次结构可以从数值上产生几乎最佳的值。将所提出的近似层次结构与现有近似结构相结合，我们可以更准确，有效地评估COP编程问题的最佳价值。
We first provide an inner-approximation hierarchy described by a sum-of-squares (SOS) constraint for the copositive (COP) cone over a general symmetric cone. The hierarchy is a generalization of that proposed by Parrilo (2000) for the usual COP cone (over a nonnegative orthant). We also discuss its dual. Second, we characterize the COP cone over a symmetric cone using the usual COP cone. By replacing the usual COP cone appearing in this characterization with the inner- or outer-approximation hierarchy provided by de Klerk and Pasechnik (2002) or Yildirim (2012), we obtain an inner- or outer-approximation hierarchy described by semidefinite but not by SOS constraints for the COP matrix cone over the direct product of a nonnegative orthant and a second-order cone. We then compare them with the existing hierarchies provided by Zuluaga et al. (2006) and Lasserre (2014). Theoretical and numerical examinations imply that we can numerically increase a depth parameter, which determines an approximation accuracy, in the approximation hierarchies derived from de Klerk and Pasechnik (2002) and Yildirim (2012), particularly when the nonnegative orthant is small. In such a case, the approximation hierarchy derived from Yildirim (2012) can yield nearly optimal values numerically. Combining the proposed approximation hierarchies with existing ones, we can evaluate the optimal value of COP programming problems more accurately and efficiently.
