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.