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• W2295929366 abstract "We propose a new proximal, path-following framework for a class of constrained convex problems. We consider settings where the nonlinear---and possibly non-smooth---objective part is endowed with a proximity operator, and the constraint set is equipped with a self-concordant barrier. Our approach relies on the following two main ideas. First, we re-parameterize the optimality condition as an auxiliary problem, such that a good initial point is available; by doing so, a family of alternative paths towards the optimum is generated. Second, we combine the proximal operator with path-following ideas to design a single-phase, proximal, path-following algorithm. Our method has several advantages. First, it allows handling non-smooth objectives via proximal operators; this avoids lifting the problem dimension in order to accommodate non-smooth components in optimization. Second, it consists of only a emph{single phase}: While the overall convergence rate of classical path-following schemes for self-concordant objectives does not suffer from the initialization phase, proximal path-following schemes undergo slow convergence, in order to obtain a good starting point cite{TranDinh2013e}. In this work, we show how to overcome this limitation in the proximal setting and prove that our scheme has the same $mathcal{O}(sqrt{nu}log(1/varepsilon))$ worst-case iteration-complexity with standard approaches cite{Nesterov2004,Nesterov1994} without requiring an initial phase, where $nu$ is the barrier parameter and $varepsilon$ is a desired accuracy. Finally, our framework allows errors in the calculation of proximal-Newton directions, without sacrificing the worst-case iteration complexity. We demonstrate the merits of our algorithm via three numerical examples, where proximal operators play a key role." @default.
• W2295929366 created "2016-06-24" @default.
• W2295929366 creator A5024280658 @default.
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• W2295929366 date "2016-03-05" @default.
• W2295929366 modified "2023-10-02" @default.
• W2295929366 title "A single-phase, proximal path-following framework" @default.
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• W2295929366 hasPublicationYear "2016" @default.
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