Geometry in nonsmooth optimization

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While nonsmooth optimization is ubiquitous across science and engineering, variational analysis - its elegant mathematical foundation - has achieved narrower practical impact than it warrants. This project attacks that deficit through "structure": not in the traditional explicit computational sense, but rather in the sense of intrinsic geometry. The investigator studies two overlapping mathematical strategies. The first uses semi-algebraic geometry as a rich and natural model for the world of concrete optimization problems. In that world, much of the technicality and pathology obscuring variational analysis for practitioners is transformed, leaving powerful stratification tools and simple access to "generic" properties in optimization. The second strategy emphasizes partial smoothness, a powerful geometric property originating from the theory of optimality conditions and sensitivity analysis, but also in perfect resonance with active-set algorithms. With these theoretical tools at hand, the investigator focuses foremost on two fresh and promising computational methods for nonsmooth optimization. The first is, counter-intuitively, just the classical BFGS method for smooth optimization, mildly adjusted for the nonsmooth world. BFGS is simple, intuitive, general-purpose, much easier to implement successfully than traditional "bundle" methods, and broadly effective in practical applications, notably in robust control. Mysteriously, BFGS always (essentially) seems to converge linearly and to identify partly smooth structure. The investigator seeks explanations. The second focal computational method is a proximal algorithm for composite optimization that is simple, versatile, and, in contrast with BFGS, well-grounded theoretically. This algorithm has proved successful on huge problems, such as compressed sensing, but is potentially slow. Partial smoothness will strengthen convergence in theory, and speed it in practice.The broader significance and importance of this project derive from the investigator's transformative goal of bridging the gulf between mathematical theory and data-driven practice in resource allocation problems beyond the reach of traditional calculus. A particularly important example is the kind of robust control engineering underlying applications like modern aircraft electronics. The investigator builds on a strong track-record of high-calibre publications, innovative scholarship and outreach, and intellectual leadership. PhD students based in Cornell's highly ranked School of ORIE (where the investigator is Director) will engage all aspects of the project, publishing and presenting at professional meetings; the investigator will engage more broadly through seminars and his award-winning teaching, as well as through graduate texts, survey articles, multidisciplinary collaboration, and prominent international lectures to broad scientific and engineering audiences.