A weakly infeasible semidefinite program (SDP) has no feasible solution, but it has approximate solutions whose constraint violation is arbitrarily small. These SDPs are ill-posed and numerically often unsolvable. They are also closely related to “bad” linear projections that map the cone of positive semidefinite matrices to a nonclosed set. We describe a simple echelon form of weakly infeasible SDPs with the following properties: (i) it is obtained by elementary row operations and congruence transformations, (ii) it makes weak infeasibility evident, and (iii) it permits us to construct any weakly infeasible SDP or bad linear projection by an elementary combinatorial algorithm. Based on our echelon form, we generate a library of computationally very difficult SDPs. Finally, we show that some SDPs in the literature are in our echelon form, for example, the SDP from the sum-of-squares relaxation of minimizing Motzkin’s famous polynomial.

弱不可行半定规划 (SDP) 没有可行解，但它具有约束违反任意小的近似解。这些 SDP 是不适定的，并且在数值上通常无法解决。它们还与将正半定矩阵的圆锥映射到非闭集的“坏”线性投影密切相关。我们描述了具有以下性质的弱不可行 SDP 的简单梯形形式：（i）它是通过基本的行操作和全等变换获得的，（ii）它使弱不可行变得明显，并且（iii）它允许我们构造任何弱不可行SDP 或基本组合算法的不良线性投影。基于我们的梯队形式，我们生成了一个计算非常困难的 SDP 库。最后，我们表明文献中的一些 SDP 是我们的梯队形式，