In semidefinite programming the solutions are symmetrical matrices (instead of vectors). Constraints are typically linear. The difference with respect to linear programming is the non negativity constraint: it is required that the matrices are positive semidefinite. Note: if the matrices are diagonal, semidefinite programming reduces to linear programming. A semidefinite programming problems is formulated in the linear space Sn of the symmetrical matrices of order n (dimension of Sn is n (n +1)/2).The space becomes a Hilbert space by introducing the inner product A o B := Tr(AB) between matrices A and B in Sn (recall Tr A := åi Aii ). Properties: A o A = 0, A o A = 0 Û A = 0, A o B =B oA , (aA) o B = a (A o B), A o B = åi Aii Bii (the last means that the inner product can be viewed as a vector product by transforming a matrix into a vector).

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