My research is in convex programming, especially in semidefinite programming, conic linear programming, integer programming, and applications of optimization. Below I list some research projects and papers that are close to my heart.

My main research area in the last few years has been on Semidefinite Programs (SDPs), some of the most exciting, and useful class of optimization problems of the last few decades.
I mainly looked at pathological SDPs. Precisely, in the duality theory of SDPs strange pathologies occur: the optimal values of primal and dual SDPs may differ and may not be attained.

Preprocessing semidefinite programs (SDPs)

A simple algorithm can remove many redundancies in SDPs, reduce their size, and often get rid of the pathological behavior, or detect infeasibility.

Understanding the pathological behavior of semidefinite programs (SDPs) and more generally, of conic linear programs (LPs).

In these few papers I give combinatorial characterizations of the pathological behaviors.I also show how to use elementary row operations (coming from Gaussian elimination) to bring semidefinite systems (or more generally, conic linear systems) into a form, so the pathological behavior becomes easy to see.

An older paper, which deals with a closely related, classical problem is

The geometry of SDPs and more generally of conic LPs (extreme points, degeneracy, etc.)

Reformulating integer programs using basis reduction

These papers show how to reformulate integer programming problems (IPs) by nearly orthogonalizing the columns. The reformulation makes many hard IPs easier, and more surprisingly, it makes most integer programs solvable by just one branch-and-bound node.

Solving a hard, previously unsolved integer program