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Invariance under affine transformation in semidefinite programming relaxation for polynomial optimization problems. (English) Zbl 1192.90138
Summary: Given a polynomial optimization problem (POP), any nonsingular affine transformation on its variable vector induces an equivalent POP. Applying J. B. Lasserre’s SDP relaxation [SIAM J. Optim. 11, No. 3, 796–817 (2001; Zbl 1010.90061)] to the original and the transformed POPs, we get 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.

MSC:
90C22 Semidefinite programming
65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
Software:
SparsePOP
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