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Feasibility and Duality

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Optimization

Part of the book series: Springer Texts in Statistics ((STS,volume 95))

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Abstract

This chapter provides a concrete introduction to several advanced topics in optimization theory. Specifying an interior feasible point is the first issue that must be faced in applying a barrier method. Given an exterior point, Dykstra’s algorithm [21, 70, 79] finds the closest point in the intersection \(\cap _{i=0}^{r-1}C_{i}\) of a finite number of closed convex sets. If C i is defined by the convex constraint \(h_{i}(\boldsymbol{x}) \leq 0\), then one obvious tactic for finding an interior point is to replace C i by the set \(C_{i}(\epsilon ) =\{ \boldsymbol{x} : h_{j}(\boldsymbol{x}) \leq -\epsilon \}\) for some small ε > 0. Projecting onto the intersection of the C i (ε) then produces an interior point.

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Lange, K. (2013). Feasibility and Duality. In: Optimization. Springer Texts in Statistics, vol 95. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5838-8_15

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