吉大計算數學系列學術會議第3期——數值優化

會議形式：網絡會議

會議日期：2020年9月25日上午 8：50-11:20

加入方式：騰訊會議ID: 870 938 043  密碼：9999

會議日程：

題目：Some Inertial Alternating Proximal(-like) Gradient Methods for a Class of Nonconvex optimization problems

報告人： 蔡邢菊 副教授

南京師範大學

摘要：We study a broad class of nonconvex nonsmooth minimization problems, whose objective function is the sum of a function of the entire variables and two nonconvex functions of each variable. For the different cases, we linearized different fart of the objective function, adopting inertial strategy to accelerate the convergence.  We also propose an inertial alternating proximal-like gradient descent algorithm for the problem with abstract constraint sets whose geometry can be captured by using the domain of kernel generating distances. This algorithm can circumvent the restrictive assumption of global Lipschitz continuity of gradient. We prove that each bounded sequence generated by these algorithms globally converge to a critical point of the problem under the assumption that the underlying functions satisfy the Kurdyka-Łojasiewicz property.

報告題目：Some Extended Proximal Point Algorithms with Applications

報告人： 韓德仁 教授

北京航空航天大學數學科學學院

摘要：The proximal point algorithm (PPA) has been widely used in convex optimization. Many algorithms fall into the framework of PPA. To guarantee the convergence of the PPA, however, existing results conventionally need to ensure the positive definiteness of the corresponding proximal measure. In this talk, we present some useful extensions of PPA.

題目：A Space Decomposition Framework for Nonlinear Optimization

報告人：張在坤 博士

香港理工大學

摘要：Space decomposition has been a popular methodology in both the community of optimization and that of numerical PDEs. Under the name of Domain Decomposition Methods, the latter community has developed highly successful techniques like Restricted Additive Schwarz and Coarse Grid. Being crucial for the performance of Domain Decomposition Methods, these techniques are however less studied in optimization. This talk presents a framework that allows us to extend these techniques to general nonlinear optimization. We establish the global convergence and convergence rate of the framework. Numerical results, although preliminary, show the Restricted Additive Schwarz and Coarse Grid (now Coarse Space) techniques can effectively accelerate space decomposition methods as they do in Domain Decomposition Methods.

This is a joint work with Serge Gratton (ENSEEIHT/INPT, University of Toulouse) and Luis Nunes Vicente (Liehigh University).

題目：Decomposition Methods for Solving Two-Stage Distributionally Robust Optimization Problems

報告人：孫海琳 教授

南京師範大學

摘要：Decomposition methods have been well studied for solving two-stage and multi-stage stochastic programming problems. In this paper, we propose an algorithmic framework based on the fundamental ideas of the methods for solving two-stage minimax distributionally robust optimization (DRO) problems where the underlying random variables take a finite number of distinct values. This is achieved by introducing nonanticipativity constraint for the first stage decision variables, rearranging the minimax problem through Lagrange decomposition and applying the well-known primal-dual hybrid gradient (PDHG) method to the new minimax problem. The algorithmic framework does not depend on specific structure of the ambiguity set. To extend the algorithm to the case that the underlying random variables are continuously distributed, we propose a discretization scheme and quantify the error arising from the discretization in terms of the optimal value and the optimal solutions when the ambiguity set is constructed through generalized prior moment conditions, the Kantorovich ball and $\phi$-divergence centred at an empirical probability distribution. Some preliminary numerical tests show the proposed decomposition algorithm featured with parallel computing performs well.