A real symmetric matrix A is copositive if \(\left\langle {Ax},{x}\right\rangle \ge 0\) for all x in the nonnegative orthant. Copositive programming gained fame when Burer showed that hard nonconvex problems can be formulated as completely-positive programs. Alas, the power of copositive programming is offset by its difficulty: simple questions like “is this matrix copositive?” have complicated answers. In 1958, Jerry Gaddum proposed a recursive procedure to check if a given matrix is copositive by solving a series of matrix games. It is easy to implement and conceptually simple. Copositivity generalizes to cones other than the nonnegative orthant. If K is a proper cone, then the linear operator L is copositive on K if \(\left\langle {L \left( {x}\right) },{x}\right\rangle \ge 0\) for all x in K. Little is known about these operators in general. We extend Gaddum’s test to self-dual and symmetric cones, thereby deducing criteria for copositivity in those settings.

如果非负正态中的所有x均为\（\ left \ langle {Ax}，{x} \ right \ rangle \ ge 0 \），则实对称矩阵A是正整数。当伯勒（Burer）证明可以将硬性非凸问题表示为完全正程序时，协同正则程序便广为人知。las，协同编程的功能被其难度所抵消：诸如“这个矩阵是否具有协同作用？”之类的简单问题。有复杂的答案。1958年，Jerry Gaddum提出了一种递归程序，通过求解一系列矩阵游戏来检查给定矩阵是否为正整数。它易于实现，概念上也很简单。共正性可推广到非负矫正剂以外的其他圆锥体。如果K是一个适当的圆锥，则线性算子大号是双正ķ如果\（\左\ langle {L \左（{X} \右）}，{X} \右\ rangle \ GE 0 \）的所有X中ķ。通常对这些运算符知之甚少。我们将Gaddum的检验扩展到自对偶和对称圆锥，从而推导了在这些情况下共正性的标准。