A striking pathology of semidefinite programs (SDPs) is illustrated by a classical example of Khachiyan: feasible solutions in SDPs may need exponential space even to write down. Such exponential size solutions are the main obstacle to solve a long standing, fundamental open problem: can we decide feasibility of SDPs in polynomial time? The consensus seems that SDPs with large size solutions are rare. However, here we prove that they are actually quite common: a linear change of variables transforms every strictly feasible SDP into a Khachiyan type SDP, in which the leading variables are large. As to ``how large", that depends on the singularity degree of a dual problem. Further, we present some SDPs coming from sum-of-squares proofs, in which large solutions appear naturally, without any change of variables. We also partially answer the question: how do we represent such large solutions in polynomial space?

翻译：半定规划（SDP）的一个著名病理学特征由Khachiyan经典例子揭示：SDP中的可行解可能需要指数级空间才能表示。这种指数规模解是解决一个长期悬而未决的基本开放问题的主要障碍：我们能否在多项式时间内判定SDP的可行性？学界共识似乎认为具有大规模解的SDP极为罕见。然而，本文证明这类解实际上相当常见：通过线性变量替换，每个严格可行的SDP均可转化为Khachiyan型SDP，其中主导变量呈现大规模特性。至于“规模多大”，这取决于对偶问题的奇异度。此外，我们给出了一些源于平方和证明的SDP实例，其中无需任何变量替换即可自然产生大规模解。本文还部分回答了以下问题：如何在多项式空间内表示这些大规模解？