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| Volume 5, Number 2, May 2009, pp. 297-312 | |||||||||||||||||||||||||||||||||||||||||||||||||
| Hayato Waki, Masakazu Muramatsu and Masakazu Kojima | |||||||||||||||||||||||||||||||||||||||||||||||||
| Key words: | |||||||||||||||||||||||||||||||||||||||||||||||||
| polynomial optimization problem, semidefinite programming relaxation, sum of squares relaxation, invariance, affine transformation, polynomial SDP | |||||||||||||||||||||||||||||||||||||||||||||||||
| Mathematices Subject Classification: 65K05, 90C22, 99C30 | |||||||||||||||||||||||||||||||||||||||||||||||||
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| Abstract: | |||
| Given a polynomial optimization problem (POP), any nonsingular affine transformation on its variable vector induces an equivalent POP. Applying Lasserre's SDP relaxation [SIAM J.Opt. 11:796--817, 2001] to the original and the transformed POPs, we have two SDPs. This paper shows that these two SDPs are isomorphic to each other under a nonsingular linear transformation, which maps the feasible region of one SDP onto that of the other isomorphically and preserves their objective values. This fact means that the SDP relaxation is invariant under any nonsingular affine transformation. | |||
| Invariance under affine transformation in semidefinite programming relaxation for polynomial optimization problems | ||