Primal-dual Interior point method
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Primal-dual interior point method is similar to barrier interior point method. Both methods are motivated in terms of perturbed KKT conditions. But there are quite big differences between them. First of all Primal-dual interior-point methods take one Newton step, And move on. and Primal-dual interior-point iterates are not necessarily feasible.
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Convex Optimization (Cambridge University Press 2004)