Best ellipsoidal relaxation to solve a nonconvex problem
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on linear algebra and arithmetic, Rabat, Morocco, 28-31 May 2001
WSEAS Transactions on Mathematics
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Search directions for primal-dual path-following methods for semidefinite programming (SDP) are proposed. These directions have the properties that (1) under certain nondegeneracy and strict complementarity assumptions, the Jacobian matrix of the associated symmetrized Newton equation has a bounded condition number along the central path in the limit as the barrier parameter $\mu$ tends to zero; and (2) the Schur complement matrix of the symmetrized Newton equation is symmetric and the cost for computing this matrix is 2mn3 + 0.5m2n2 flops, where n and m are the dimension of the matrix and vector variables of the semidefinite program, respectively. These two properties imply that a path-following method using the proposed directions can achieve the high accuracy typically attained by methods employing the direction proposed by Alizadeh, Haeberly, and Overton (currently the best search direction in terms of accuracy), but each iteration requires at most half the amount of flops (to leading order).