## Adrian Lewis presents invited lecture to the International Congress of Mathematicians

The International Congress of Mathematicians, held in August 2014 in Seoul, Korea, is the world’s premier mathematics conference. Several prominent prizes, including the Fields Medals, are awarded at the opening of the eight-day event, and many famous talks – including Hilbert’s influential listing of 23 seminal mathematical problems – have been delivered there. Invited lectures are among the mathematical community’s greatest honors.

ORIE Professor Adrian Lewis was invited to speak at this year’s conference, in the section devoted to control theory and optimization. He presented a fresh unifying perspective on several mathematical areas, under the title “*Nonsmooth optimization: conditioning, convergence and semi-algebraic models*.” His talk covered a “web” of related ideas in optimization, computational algorithms, robust control, calculus, geometry, matrix analysis, and diverse applications. He illustrated his approach with intuitive geometric constructions and showed computational examples.

Lewis’s theme – “computational inversion” – drew an analogy between a rich variety of mathematical problems and solving simple equations, in which one seeks numerical solutions from given data. As a simple example, think of inverting the quadratic equation , that is, finding the two real solutions x (when they exist) in terms of the data y by completing the square, taking square roots, and so forth.. More difficult systems of equations or relationships (like linear programming) demand MORE sophisticated inversion algorithms.

Computational specialists often ask three key questions about any particular inversion example: How sensitive is the answer to errors in the data? How close is the example to being unsolvable? How fast can we solve it? Lewis’s central topic, influenced by work of ORIE Professor Jim Renegar (an invitee to an earlier Congress), was to unify the answers to these questions.

Lewis’s lecture described his work with many co-authors, building on this unifying idea to illuminate diverse mathematical problems, especially active set methods in constrained optimization, and creating algorithms ranging from classical alternating projection ideas to contemporary machine learning. Along the way, he discussed a highly (although so far inexplicably) successful reincarnation of a popular method for classical smooth optimization due to Broyden, Fletcher, Goldfarb ‘63 and Shanno, retooled for challenging nonsmooth optimization problems.

In his talk, Lewis reminisced about attending the 1990 Congress in Kyoto, Japan, not long after receiving his PhD in Engineering from Cambridge University, where he had previously received a BA in Mathematics. He recalled admiring the invited speakers on the stage (including ORIE faculty members Jim Renegar and Eva Tardos), never imagining that he would be among them twenty-four years later.

Lewis joined ORIE in 2004 and was Director of the School from 2010 to 2013.