Linear and Nonlinear Optimization
This course systematically explores foundational theories and classical algorithms in linear and nonlinear mathematical optimization from both theoretical and practical perspectives. The course is structured into six detailed chapters: The first chapter introduces optimization problems and modeling techniques, providing a concise overview of mathematical foundations essential to optimization, such as linear algebra, calculus, and matrix theory. The second chapter thoroughly examines convex analysis, covering concepts like convex sets, convex functions, and extending to Difference-of-Convex (DC) structures. Chapter three presents duality theory, focusing on Lagrangian duality and saddle point theory, and their application in optimization. The fourth chapter explores optimality conditions, outlining necessary and sufficient conditions crucial for solving both constrained and unconstrained optimization problems. Chapter five covers linear optimization, highlighting fundamental theoretical aspects and practical algorithms like the simplex method. The final chapter delves into nonlinear optimization, including both convex and non-convex optimization algorithms such as gradient-based, Newton-type, and DC algorithms. Throughout the course, practical training sessions using software tools such as MATLAB and CPLEX are provided to enable students to effectively bridge theoretical knowledge and practical application in solving real-world optimization problems.
Lecturer
Date
15th September ~ 11th December, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Thursday | 09:50 - 11:25 | Qiuzhen | ZOOM 08 | 787 662 9899 | BIMSA |
Prerequisite
Linear algebra, Analysis, Calculus
Syllabus
1. Optimization problems and modeling
2. Mathematical background
3. Convex analysis
4. Lagrangian duality theory
5. Optimality conditions
6. Linear programming
7. Convex optimization theory and algorithms
8. Non-convex optimization theory and algorithms
9. Optimization modeling software and solvers
2. Mathematical background
3. Convex analysis
4. Lagrangian duality theory
5. Optimality conditions
6. Linear programming
7. Convex optimization theory and algorithms
8. Non-convex optimization theory and algorithms
9. Optimization modeling software and solvers
Reference
1. 最优化理论和算法（法文版）– Y.S. Niu and H.J. Ji
2. Convex Optimization – S. Boyd and L. Vandenberghe
3. Nonlinear Programming – D.P. Bertsekas
4. Lectures on Convex Optimization – Y. Nesterov
5. First-Order Methods in Optimization – A. Beck
6. Linear and Nonlinear Programming – D.G. Luenberger and Y.Y. Ye
2. Convex Optimization – S. Boyd and L. Vandenberghe
3. Nonlinear Programming – D.P. Bertsekas
4. Lectures on Convex Optimization – Y. Nesterov
5. First-Order Methods in Optimization – A. Beck
6. Linear and Nonlinear Programming – D.G. Luenberger and Y.Y. Ye
Audience
Undergraduate
, Advanced Undergraduate
, Graduate
Video Public
No
Notes Public
No
Language
Chinese
, English
Lecturer Intro
Yi-Shuai Niu is an Associate Professor at the Beijing Institute of Mathematical Sciences and Applications (BIMSA), specializing in optimization, scientific computing, machine learning, and computer science. He also holds a dual appointment at Tsinghua University (Qiuzhen College), where he teaches optimization- and AI-related courses and supervises graduate and undergraduate research. Before joining BIMSA, he was a Research Fellow at The Hong Kong Polytechnic University (2021–2022) and an Associate Professor at Shanghai Jiao Tong University (2014–2021), where he founded the Optimization and Interdisciplinary Research Group and held concurrent appointments at the SJTU-ParisTech Elite Institute of Technology and the School of Mathematical Sciences. His earlier positions included Postdoctoral Fellow at the University of Paris 6 (2013–2014), Junior Researcher at CNRS and Stanford University (2010–2012), and Lecturer at INSA Rouen (2007–2010). He received his Ph.D. in Mathematics–Optimization in 2010 and master’s degrees in "Pure and Applied Mathematics" and "Engineering Mathematics" in 2006.
His research focuses on optimization theory, machine learning, high-performance computing (HPC), and scientific software, with applications in natural language processing, autonomous driving, finance, image processing, turbulent combustion, polymer science, quantum computing, and plasma physics. He develops new theories and algorithms for large-scale nonconvex and nonsmooth optimization, together with efficient HPC-based solvers and scientific computing packages. He has developed more than 36 software packages and published over 40 papers in leading journals and conference proceedings, including the SIAM Journal on Optimization, Journal of Scientific Computing, and Combustion and Flame. He has served as PI of 7 research grants, including an NSFC Key Project, and as a core member of 5 international collaborative projects.
His honors include the Beijing High-Level Overseas Talent Programs, core membership in the Beijing Strategic Scientist Program, the First Prize of the 2017 Shanghai Teaching Achievement Award, and First Prizes of the Shanghai Jiao Tong University Teaching Achievement Awards in 2016 and 2017; he has received 17 MCM/ICM awards, including the 2017 INFORMS Best Paper Award. In 2026, he was selected for the International Congress of Basic Science (ICBS) Innovation Award and named a "Li Bing Engineering Innovation Scholar".
His research focuses on optimization theory, machine learning, high-performance computing (HPC), and scientific software, with applications in natural language processing, autonomous driving, finance, image processing, turbulent combustion, polymer science, quantum computing, and plasma physics. He develops new theories and algorithms for large-scale nonconvex and nonsmooth optimization, together with efficient HPC-based solvers and scientific computing packages. He has developed more than 36 software packages and published over 40 papers in leading journals and conference proceedings, including the SIAM Journal on Optimization, Journal of Scientific Computing, and Combustion and Flame. He has served as PI of 7 research grants, including an NSFC Key Project, and as a core member of 5 international collaborative projects.
His honors include the Beijing High-Level Overseas Talent Programs, core membership in the Beijing Strategic Scientist Program, the First Prize of the 2017 Shanghai Teaching Achievement Award, and First Prizes of the Shanghai Jiao Tong University Teaching Achievement Awards in 2016 and 2017; he has received 17 MCM/ICM awards, including the 2017 INFORMS Best Paper Award. In 2026, he was selected for the International Congress of Basic Science (ICBS) Innovation Award and named a "Li Bing Engineering Innovation Scholar".