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).

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).

Faculty