# Research and selected publications

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.

- Sieve-SDP: a simple facial reduction algorithm to preprocess semidefinite programs Y. Zhu, G. Pataki, Q. Tran-Dinh

*Submitted*

** 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.

- Bad semidefinite programs with short proofs, and the closedness of the linear image of the semidefinite cone G. Pataki

*Submitted*

A talk in 2017

**Remark: a much simplified treatment of the SDP results of the “Bad SDP” paper; that paper, however, addresses general conic linear programs** - Exact duals and short certificates of infeasibility and weak infeasibility in conic linear programming , M. Liu, G. Pataki,

*To appear, Mathematical Programming, series A***DOI**

A talk, Aug 2016 - Bad semidefinite programs: they all look the same G. Pataki

*SIAM Journal on Optimization, 2017, 27(3), 146–172, 2017***DOI**

A talk in 2014 - Exact duality in semidefinite programming based on elementary reformulations , M. Liu, G. Pataki,

*SIAM Journal on Optimization, 25(3), 1441–1454, 2015***DOI**

Talk at Tamas Terlaky’s birthday conference June 2015

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

- On the Closedness of the Linear Image of a Closed Convex Cone , G. Pataki

*Mathematics of Operations Research. 32(2), 395-412, 2007***DOI**

Talk at Dalhousie University, January 2005

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

- The Geometry of Semidefinite Programming G. Pataki,

*In the The Handbook of Semidefinite Programming, Kluwer, 2000*

Talk at Cornell University, 1998 - On the Rank of Extreme Matrices in Semidefinite Programs and the

Multiplicity of Optimal Eigenvalues, G. Pataki

*Mathematics of Operations Research, 23 (2), 339-358, 1998***DOI**

** 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.

*Basis Reduction, and the Complexity of Branch-and-Bound, G. Pataki, M. Tural, E. B. Wong*

*2010 ACM-SIAM Symposium on Discrete Algorithms (SODA 10)***DOI**

Talk at the Combinatorial Optimization workshop in Aussois, 2011- Column Basis Reduction and Decomposable Knapsack Problems, B. Krishnamoorthy and G. Pataki,

*Discrete Optimization, 6(3), August 2009, 242-270***DOI**

A talk, January 2005

** Solving a hard, previously unsolved integer program **

- Solving the seymour problem, M. C. Ferris, G. Pataki and S. Schmieta

*Optima, 66:1-7, 2001.*

Talk at ISMP 2000 (given by Stefan Schmieta)