Kronecker Product Constraints for Semidefinite Optimization
نویسنده
چکیده
We consider semidefinite optimization problems that include constraints of the form G(x) 0 and H(x) 0, where the components of the symmetric matrices G(·) and H(·) are affine functions of x ∈ Rn. In such a case we obtain a new constraint K(x,X) 0, where the components of K(·, ·) are affine functions of x and X, and X is an n×n matrix that is a relaxation of xxT . The constraint K(x,X) 0 is based on the fact that G(x)⊗H(x) 0, where ⊗ denotes the Kronecker product. This construction of a constraint based on the Kronecker product generalizes the construction of an RLT constraint from two linear inequality constraints, and also the construction of an SOC-RLT constraint from one linear inequality constraint and a second-order cone constraint. We show how the Kronecker product constraint obtained from two second-order cone constraints can be efficiently used to computationally strengthen the semidefinite programming relaxation of the two-trust-region subproblem.
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