As a classic example of Khachiyan shows, some semidefinite programs (SDPs) have solutions whose size -- the number of bits necessary to describe them -- is exponential in the size of the input. Exponential size solutions are the main obstacle to solve a long standing open problem: can we decide feasibility of SDPs in polynomial time? We prove that large solutions are actually quite common in SDPs: 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 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?

中文翻译：

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半定规划中指数大小的解决方案是如何产生的？

正如Khachiyan的经典示例所示，某些半定程序（SDP）的解决方案的大小（描述它们的必要位数）在输入大小上呈指数关系。指数大小的解决方案是解决长期存在的开放问题的主要障碍：我们能否确定多项式时间内SDP的可行性？我们证明了大型解决方案实际上在SDP中非常普遍：变量的线性变化将每个严格可行的SDP转换为Khachiyan类型的SDP，其中前导变量很大。至于“有多大”，这取决于对偶问题的奇异程度。此外，我们介绍了一些SDP，其中大型解决方案自然而然地出现了，没有任何变量的变化。我们还部分回答了这个问题：我们如何在多项式空间中表示如此大的解？