# Optimization

# Optimization

The Optimization Group was founded in 1946 as a subgroup of Operations Research Program at UNC-Chapel Hill. Our research develops advanced theory and algorithms to analyze and solve optimization problems arising from applications.

Our research covers a wide range of topics, such as convex and variational analysis, semidefinite programming, convex and nonconvex programming, complementarity problems and variational inequalities, integer programming, stochastic optimization, and optimal control. By exploiting the fundamental structure in optimization problems, we have developed new computational and analytical methods and brought new insights into important classes of problems.

We have also applied optimization techniques in a broad range of areas, including engineering, economics, physical, chemical, and biological sciences, business and management, data science and machine learning, transportation science, and social sciences.

## Research topics

**Theory and methodology for nonlinear optimization**

- Semidefinite programming and conic programming
- Polynomial and semi-algebraic optimization
- Symbolic computation, applied topology, and real algebraic geometry
- Integer programming
- Primal-dual methods for large-scale convex optimization
- Self-concordant convex optimization
- Numerical methods for nonlinear optimization
- Sequential convex programming and Gauss-Newton methods
- Optimality and constant rank constraint qualification in optimization

**Variational inequalities and equilibrium problems**

- Optimization under uncertainty; stochastic optimization and equilibria
- Sensitivity analysis for variational inequalities and equilibrium problems
- confidence regions and intervals for stochastic variational inequalities
- Sensitivity analysis of traffic user equilibria
- Inference for sparse penalized statistical regression

**Applications of Optimization**

- Optimal control, linear and nonlinear model predictive control (MPC)
- Linear feedback controller design
- Machine learning and statistics
- Image and signal processing, and compressive sensing
- Operations Research

## Special Topics Courses

- Semidefinite Programming and Integer Programming (Pataki)
- Large-Scale Optimization in Machine Learning (Tran-Dinh)
- Optimization for Machine Learning and Data Analysis (Tran-Dinh)
- Topics on Numerical Methods for Modern Optimization in Data Analysis (Tran-Dinh)
- Topics in Optimization, Integer Programming and Semidefinite Programming (Pataki)
- Topics in Modern Convex Optimization (Tran-Dinh)

## Representative Publications

S. Basu & A. Mohammad-Nezhad

On the complexity of analyticity in semidefinite optimization

*Advances in Applied Mathematics* (2024).

S. Basu & A. Mohammad-Nezhad

On the central path of semidefinite optimization: degree and convergence rate

*SIAM Journal on Applied Algebra and Geometry* (2022).

F. E. Curtis, M. J. O’Neill & D. P. Robinson

Worst-case complexity of an SQP method for nonlinear equality constrained stochastic optimization

*Mathematical Programming* (2023).

Q. Tran-Dinh, N. H. Pham, D. T. Phan & L. M. Nguyen

A Hybrid Stochastic Optimization Framework for Stochastic Composite Nonconvex Optimization

*Mathematical Programming* 181:1-67 (2021).

T. Sun & Q. Tran-Dinh

Generalized Self-Concordant Functions: A Recipe for Newton-Type Methods

*Mathematical Programming* 178:145–213 (2019).

G. Pataki

Characterizing Bad Semidefinite Programs: Normal Forms and Short Proofs

*SIAM Review* 61(4):839–859 (2019).

Z. Qi, Y. Cui, Y. Liu & J. S. Pang

Estimation of Individualized Decision Rules Based on An Optimized Covariate-dependent Equivalent of Random Outcomes

*SIAM Journal on Optimization* 29(3):2337–2362 (2019).

C. Qian, Q. Tran-Dinh, S. Fu, C. Zou & Y. Liu

Robust Multicategory Support Matrix Machine

*Mathematical Programming* 176(1):429-463 (2019).

C. Zhang, M. Pham, S. Fu & Y. Liu

Robust Multicategory Support Vector Machines using Difference Convex Algorithm

*Mathematical Programming* 169(1):277-305 (2018).

Q. Tran-Dinh, O. Fercoq & V. Cevher

A Smooth Primal-Dual Optimization Framework for Nonsmooth Composite Convex Minimization

*SIAM Journal on Optimization* 28(1):96–134 (2018).

M. Liu & G. Pataki

Exact Duals and Short Certificates of Infeasibility and Weak Infeasibility in Conic Linear Programming.

*Mathematical Programming* 167:435-480 (2018).

G. Pataki

Bad Semidefinite Programs: They All Look the Same.

*SIAM Journal on Optimization* 27(3):146-172 (2017).

Q. Tran-Dinh, A. Kyrillidis & V. Cevher

Composite Self-Concordant Minimization

*Journal of Machine Learning Research* 16:71−416 (2015).

G. Pataki

On the Rank of Extreme Matrices in Semidefinite Programs and the Multiplicity of Optimal Eigenvalues Sets

*Mathematics of Operations Research* 23(2):339-358 (1998).