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699-approximation algorithm for Max-Bisection by Ye Y. PDF

By Ye Y.

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Since the centrality conditions have a unique solution, it follows that in this interval. The implicit function theorem also supplies an expression for the tangential direction to the central path. 9). The tangential direction is the direction used by all primal-dual path-following interior point methods if the current iterate is on the central path. ) Halická [77] has recently shown that the central path can be analytically extended (see the definition in Appendix D) to in the case of strict complementarity.

Proof: Let be an arbitrary optimal pair. 1) and we obtain Since , dividing both sides by yields for all . 13) are nonnegative. We derive from this that and are maximally complementary. Below we give the derivation for ; the derivation for is similar and is therefore omitted. 14) yields 50 Aspects of Semidefinite Programming The last inequality implies Letting go to infinity we obtain where denotes the column of we have whenever and the diagonal element of This implies . Thus since where is the symmetric square root factor of .

Now let be an orthogonal matrix whose columns form an orthonormal basis for the subspace For example, if we can choose as the submatrix of Q obtained by taking the columns corresponding to positive eigenvalues. Note that the spaces and are orthogonal to each other. 4 Thus we also define the matrices and as orthogonal matrices such that and The matrices and are not uniquely defined, of course, but since an orthonormal basis of a given subspace of is unique up to a rotation, we have the following.

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699-approximation algorithm for Max-Bisection by Ye Y.

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