# “运筹学基础与发展论坛”系列活动

DOOR #1:“变分分析--基础理论与前沿进展”（线上）课程 2021/10/26-2021/12/10

### 课程专栏五（11.22 -- 11.28）

Jane J. Ye，加拿大维多利亚大学教授，1982年本科毕业于厦门大学，分别于1986年和1990年获达尔豪斯大学MBA和应用数学博士学位。1990-1992年师从非光滑分析奠基人Francis Clarke在蒙特利尔大学数学研究中心做博士后研究，1992年加入维多利亚大学，2002年晋升正教授。主要研究领域为最优化与最优控制理论，变分分析和双层优化及其应用。现任最优化领域权威期刊SIAM J. Optim.，Math. Oper. Res. Set-Valued Var. Anal. 编委，2015年获加拿大数学学会颁发的Krieger-Nelson Prize

Topic: BILEVEL PROGRAM AND MATHEMATICAL PROGRAM WITH EQUILIBRIUM CONSTRAINTS VIA VARIATIONAL ANALYSIS

This course consists of four lectures devoted to using variational analysis as a tool to study bilevel programs and mathematical program with equilibrium constraints.

Optimality conditions for the mathematical program with equilibrium constraints via variational analysis

20211123上午1000--1130

In this lecture, I will explain why the mathematical program with equilibrium constraints is an intrinsic nonconvex and nonsmooth optimization problem even when all defining functions are smooth. I will then discuss how we could use the tool of variational analysis to derive various optimality conditions.

Bilevel programs: introduction, reformulations, and partial calmness condition

20211125上午1000--1130

In this lecture, I will first give an introduction to bilevel programs. Then I will discuss difficulties in studying bilevel programs and review various reformulations for bilevel programs. Finally, I will discuss the concept of partial calmness condition.

Bilevel programs: directional constraint qualifications and necessary optimality conditions

20211126上午1000--1130

This lecture is devoted to constraint qualifications and KKT conditions for bilevel programs. Directional optimality condition is sharper than the classical KKT condition while directional constraint qualification is weaker than the classical nondirectional one. I will explain how we could obtain verifiable directional constraint qualification and directional necessary optimality conditions for bilevel programs.

DC bilevel algorithms with applications in hyperparameter optimization

20211127上午1000--1130

In this lecture, I will first explain how one can use convex analysis as a tool to design algorithms for solving a class of nonsmooth bilevel program which is a class of bilevel programs where the upper level objective function is a difference of (nonsmooth) convex functions and the lower level programs are fully convex (but nonsmooth). Then I will discuss its applications in hyperparameter selection in machine learning.