How do exponential size solutions arise in semidefinite programming?

02/26/2021
by   Gábor Pataki, et al.
0

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?

READ FULL TEXT

Please sign up or login with your details

Forgot password? Click here to reset