On the Complexity of Semide nite Programs

We show that the feasibility of a system of m linear inequalities over the cone of symmetric positive semideenite matrices of order n can be tested in mn O(minfm;n 2 g) arithmetic operations with ln O(minfm;n 2 g)-bit numbers, where l is the maximum binary size of the input coeecients. We also show that any feasible system of dimension (m; n) has a solution X such that log kXk ln O(minfm;n 2 g) .

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