## Solving High-dimensional Shape-constrained Convex Regression Problems

**主讲人：** Defeng Sun(The Hong Kong Polytechnic University)

**活动时间：** 从 2021-08-11 10:00 到 11:00

**场地：** Online

Tencent Meeting ID: 422 996 976 https://meeting.tencent.com/dm/rWTtSofYwzzl

Abstract: Shape-constrained convex regression problem deals with fitting a convex function to the observed data, where additional constraints are imposed, such as component-wise monotonicity and uniform Lipschitz continuity. In this talk we provide a unified framework for computing the least squares estimator of a multivariate shape-constrained convex regression function in R^d. We prove that the least squares estimator is computable via solving a constrained convex quadratic programming (QP) problem with (n+1)d variables, n(n-1)linear inequality constraints and n possibly non-polyhedral inequality constraints, where n is the number of data points. To efficiently solve the generally very large-scale convex QP, we design a proximal augmented Lagrangian method (pALM) whose subproblems are solved by the semismooth Newton method (SSN). To further accelerate the computation when n is huge, we design a practical implementation of the constraint generation method such that each reduced problem is efficiently solved by our proposed pALM. Comprehensive numerical experiments, including those in the pricing of basket options and estimation of production functions in economics, demonstrate that our proposed pALM outperforms the state-of-the-art algorithms, and the proposed acceleration technique further shortens the computation time by a large margin.

Biodata: Professor Defeng Sun is currently Chair Professor of Applied Optimization and Operations Research at the Hong Kong Polytechnic University and the President of the Hong Kong Mathematical Society. He mainly publishes in non-convex continuous optimization and machine learning. Together with Professor Kim-Chuan Toh and Dr Liuqin Yang, he was awarded the triennial 2018 Beale--Orchard-Hays Prize for Excellence in Computational Mathematical Programming by the Mathematical Optimization Society. He served as editor-in-chief of Asia-Pacific Journal of Operational Research from 2011 to 2013 and he now serves as associate editor of Mathematical Programming, SIAM Journal on Optimization, Journal of Optimization Theory and Applications, Journal of the Operations Research Society of China, Journal of Computational Mathematics, and Science China: Mathematics. In 2020, he was elected as a Fellow of the societies CSIAM and SIAM.