Conic programming, especially semidefinite programming, has applications in control; design of stable structures; approximation algorithms in combinatorial optimization; planning under uncertainty including robust optimization; and more recently in polynomial optimization and lift-and-project schemes for solving integer and nonconvex problems.
My current research focus is on developing conic interior point based decomposition methods aimed at solving large scale structured semidefinite programs (SDPs) (say with a block angular structure) that arise in practice. One can also exploit the sparsity and/or the symmetry in the underlying SDP to preprocess it into an equivalent SDP with a block-diagonal/block-angular structure. Our approach then solves such a structured SDP using existing interior point methods (IPMs); in an iterative fashion between a coordinating master problem (mixed conic problem over linear, second-order, and smaller semidefinite cones); and decomposed and distributed subproblems (smaller SDPs) in a parallel and distributed high performance computing environment.
Recently, I have started working on SOS (sum of squares) and moment techniques for solving polynomial optimization problems. These approaches generate a hierarchy of semidefinite approximations whose objective values converge (under certain assumptions) to a global optimal solution of the polynomial program. Typically, the polynomials are sparse (few nonzero coefficients) or possess some underlying symmetry (invariant under the action of a group). This allows one to generate structured (block angular/block diagonal) semidefinite programming approximations to polynomial programs and one can solve these SDPs using a parallel conic interior point decomposition scheme.
I am also interested in incorporating the conic decomposition approach in the pricing phase of a semidefinite programming based conic branch-cut-price framework for solving mixed-integer and nonconvex problems arising in industry.
Other interests include development of active set and simplex-like approaches for solving conic problems, including second-order, and semidefinite programming; and using these algorithms to warm-start mixed integer conic problems after branching or addition of cutting planes.
My publications, preprints, and recent talks can be downloaded from my publications page. A sampling of recent talks is also available.
I will be teaching a course MA 796S/OR 791K: Convex Programming and Interior Point Methods in Fall 2007 that will touch on some of these research topics.