Variational Analysis for Practical Optimization

Cornell University, Ithaca NY

Investigators

Abstract

Lewis DMS-0806057 The investigator and his students study the interplay between variational analysis, intuitive, possibly randomized, general-purpose nonsmooth optimization algorithms, and applications to concrete models, particularly in robust control. A paradigm, originating with Demmel in the 1980's, relating conditioning, ill-posedness, and computational speed, inspires the investigator's thinking. The study of matrix pseudospectra powerfully illustrates the general technique: fundamental indicators of transient dynamical behavior, pseudospectra have a rich structure, blending complex and matrix analysis, semi-algebraic geometry, and numerical linear algebra, all ingredients in pseudospectral optimization. Semi-algebraic geometry in particular is developing into a fundamental tool not just in applications such as pseudospectra, but throughout variational analysis. The investigator uses this approach, for example, to study novel structural tools such as "partial smoothness", thereby analyzing convergence rates observed in computational practice. Fundamental to the investigator's approach is a complex blend of variational analysis, classical mathematics, numerical computation, and applied modeling. Often the designer of a complex engineering system varies certain features in order to optimize some aspect of system performance. A simple example is a shock-absorber: by varying the grade of lubricant it contains, the manufacturer can modify how stiff a suspension system feels. Finding optimal choices for damping vibrations in such a physical, electronic, communications or information system is "nonsmooth": choices are very sensitive to slight changes in the system, one reason they seem hard to compute. The investigator facilitates this important but challenging optimization process by blending mathematical modeling and analysis with scientific computation. He disseminates his research energetically through expository and technical publications, high-profile lectures, and online, to diverse international scientific and engineering audiences. As an integral part of the project, the investigator's PhD students develop crucial skills bridging divisions between sophisticated mathematical theory and the many practical applications of nonsmooth optimization.