# Research and selected publications

My research is in convex programming, especially in semidefinite programming, and conic linear programming. A few years back I also did research in integer programming. I also have several papers on applications of optimization.

My main research area, which is closest to my heart, is on Semidefinite Programs (SDPs), some of the most exciting, and useful class of optimization problems of the last few decades.

The goal is to produce ** easy-to-verify, combinatorial certificates ** for SDPs: for example, to verify

- infeasibility,
- weak infeasibility: when an SDP is infeasible, but within zero distance to the set of feasible instances. Weak infeasibility does not happen in linear programming and it leads to very challenging SDPs.
- Pathological behavior of semidefinite systems: why essentially all “bad” SDPs in the literature look strikingly similar.
- Positive duality gaps — again a fascinating behavior that leads to very difficult SDPs and cannot happen in linear programming.
- Exponential size solutions — these are impossible to even write down in a reasonable amount of space.

Interestingly, we can use mostly elementary row operations (coming from Gaussian elimination) to bring SDPs (more generally, conic linear systems) into a form, so the infeasibility, weak infeasibility, etc. become easy to see.

The structural insights that we obtain can be used to preprocess SDPs and to create difficult SDP instance libraries.

- How do exponential size solutions arise in semidefinite programming? G. Pataki, A. Touzov

*To appear, SIAM Journal on Optimization*A talk

Poster talk at the MIP 2021 conference, May 24, 2021**2nd prize to Alex Touzov** - A simplified treatment of Ramana’s exact dual for semidefinite programming B. Lourenco, G. Pataki,

*Optimization Letters (2022): 1-25.* - An echelon form of weakly infeasible semidefinite programs and bad projections of the psd cone G. Pataki, A. Touzov

*Foundations of Computational Mathematics, 2022, 1–38.*A talk - On positive duality gaps in semidefinite programming G. Pataki

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

*Mathematical Programming Computation (2019)*A talk**DOI**

Poster talk at the 2018 Princeton Optimization Day, sept 2018.**First prize to Yuzixuan Zhu in the Algorithms category.** - Characterizing bad semidefinite programs: normal forms and short proofs G. Pataki

*SIAM Review (2019)*A talk**DOI** - Exact duals and short certificates of infeasibility and weak infeasibility in conic linear programming , M. Liu, G. Pataki,

*Mathematical Programming, Ser. A (2018) 167:435–480*A talk**DOI** - Bad semidefinite programs: they all look the same G. Pataki

*SIAM Journal on Optimization, 2017, 27(3), 146–172, 2017*A talk**DOI** - Exact duality in semidefinite programming based on elementary reformulations , M. Liu, G. Pataki,

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

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*A talk**DOI**

** 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*A talk - 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)*A talk**DOI**- Column Basis Reduction and Decomposable Knapsack Problems, B. Krishnamoorthy and G. Pataki,

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

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

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

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