Investigating interior-point methods in optimization:
This is a class of efficient algorithms developed in the last fifteen years, initially for linear programming and more recently for a wide range of convex programming problems, including semidefinite programming. With K.-C. Toh and R. H. Tutuncu, I continued my work on an efficient code for solving semidefinite programming problems (SDP). In SDP, the variable to be optimized is not a vector but a symmetric matrix, with applications in approximations for combinatorial optimization problems and in control theory among others. The code is also capable of handling second order cones, as arise in quadratic optimization. The new version is available on the web for users. Graduate student Alper Yildirim completed his dissertation with me on sensitivity analysis in interior-point methods, both for linear and for semidefinite programming. I spent the fall semester visiting the Department of Economics at Yale and the T.J. Watson Research Center of IBM, working on a problem in mathematical economics (consumer demand theory) (with Ana Fostel and Herb Scarf) and increased accuracy in search direction computation in interior-point methods (with Katya Scheinberg). The spring was spent in Ithaca, with shorter visits to Carnegie-Mellon University and the Fields Institute for Research in Mathematical Sciences in Toronto, working on detecting infeasibility and unboundedness, efficient handling of fixed and free variables (with Reha Tütüncü) and an application of conic optimization to data classification problems (with Steve Marron).