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.

15th September ~ 4th December, 2025

Weekday	Time	Venue	Online	ID	Password
Monday,Thursday	09:50 - 11:25	Qiuzhen	ZOOM 08	787 662 9899	BIMSA

Linear algebra, Analysis, Calculus

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

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

Undergraduate , Advanced Undergraduate , Graduate

No

Chinese , English

Yi-Shuai Niu, a tenured Associate Professor of Mathematics at Beijing Institute of Mathematical Sciences and Applications (BIMSA), specialized in Optimization, Scientific Computing, Machine Learning, and Computer Sciences. Before joining BIMSA in October 2023, he was a research fellow at the Hong Kong Polytechnic University (2021-2022); an associate professor at Shanghai Jiao Tong University (2014-2021), where he led the “Optimization and Interdisciplinary Research Group” and double-appointed at the ParisTech Elite Institute of Technology and the School of Mathematical Sciences. His earlier roles include postdoc at the University of Paris 6 (2013-2014) and junior researcher both at the French National Center for Scientific Research (CNRS) and Stanford University (2010-2012). He was also a lecturer at the National Institute of Applied Sciences (INSA) of Rouen (2007-2010) in France, where he earned a Ph.D. in Mathematics-Optimization in 2010 and double Masters in Pure and Applied Mathematics and Genie Mathematics in 2006. His research covers a wide range of applied mathematics, with a spotlight on optimization theory, machine learning, high-performance computing, and software development. His works span various interdisciplinary applications including: machine learning, natural language processing, self-driving car, finance, image processing, turbulent combustion, polymer science, quantum chemistry and computing, and plasma physics. His contributions encompass fundamental research, emphasizing novel algorithms for large-scale nonconvex and nonsmooth problems, and practical implementations, focusing on efficient optimization solvers and scientific computing packages using high-performance computing techniques. He developed more than 33 pieces of software and published about 30 articles in prestigious journals and conferences (including SIAM Journal on Optimization, Journal of Scientific Computing, Combustion and Flames, Applied Mathematics and Computation). He was PI of 5 research grants and members of 5 joint international research projects. He was awarded of shanghai teaching achievement award (First prize) in 2017, two outstanding teaching awards (First prize) at Shanghai Jiao Tong University in 2016 and 2017 respectively, as well as 17 awards in international contests of mathematics MCM/ICM (including the INFORMS best paper award in 2017).