Integral Quadratic Constraints, For Delayed Nonlinear And Parameter

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Download PDF Abstract: This manuscript develops a new framework to analyze and design iterativeoptimization algorithms built on the notion of Integral Quadratic Constraints(IQC) from robust control theory. IQCs provide sufficient conditions for thestability of complicated interconnected systems, and these conditions can bechecked by semidefinite programming. We discuss how to adapt IQC theory tostudy optimization algorithms, proving new inequalities about convex functionsand providing a version of IQC theory adapted for use by optimizationresearchers. Using these inequalities, we derive numerical upper bounds onconvergence rates for the gradient method, the heavy-ball method, Nesterov”saccelerated method, and related variants by solving small, simple semidefiniteprogramming problems. We also briefly show how these techniques can be used tosearch for optimization algorithms with desired performance characteristics,establishing a new methodology for algorithm design.

Comments: The previous version of this paper quoted the wrong rate of Nesterov”s optimal method when applied to strongly convex functions. With this correction, our bounds are now slightly better than those previously derived for Nesterov”s method
Subjects: Optimization and Control (math.OC); Systems and Control (eess.SY); Numerical Analysis (math.NA)
DOI: 10.1137/15M1009597
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Submission history

From: Laurent Lessard Fri, 15 Aug 2014 17:49:50 UTC (1,134 KB) Tue, 23 Dec 2014 02:39:49 UTC (1,139 KB) Wed, 24 Dec 2014 10:21:26 UTC (1,139 KB) Wed, 29 Jul 2015 01:56:29 UTC (1,131 KB) Thu, 1 Oct 2015 06:13:01 UTC (1,131 KB) Sun, 25 Oct 2015 20:54:33 UTC (1,094 KB)Wed, 28 Oct 2015 19:46:52 UTC (1,094 KB)

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