 Smooth Convex OptimizationYurii Nesterov
Chapter Chapter 2 in Lectures on Convex Optimization, 2018, pp 59-137 from Springer Abstract: Abstract In this chapter, we study the complexity of solving optimization problems formed by differentiable convex components. We start by establishing the main properties of such functions and deriving the lower complexity bounds, which are valid for all natural optimization methods. After that, we prove the worst-case performance guarantees for the Gradient Method. Since these bounds are quite far from the lower complexity bounds, we develop a special technique, based on the notion of estimating sequences, which allows us to justify the Fast Gradient Methods. These methods appear to be optimal for smooth convex problems. We also obtain performance guarantees for these methods targeting on generating points with small norm of the gradient. In order to treat problems with set constraints, we introduce the notion of a Gradient Mapping. This allows an automatic extension of methods for unconstrained minimization to the constrained case. In the last section, we consider methods for solving smooth optimization problems, defined by several functional components. Date: 2018 There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-319-91578-4_2 Ordering information: This item can be ordered from DOI: 10.1007/978-3-319-91578-4_2 Access Statistics for this chapter More chapters in Springer Optimization and Its Applications from Springer |
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