Session: Stochastic Optimization
Chair: Honghao Zhang
Cluster: nan
Talk 1: Multi-cut stochastic approximation methods for solving stochastic convex composite optimization
Speaker: Honghao Zhang
Abstract: The development of a multi-cut stochastic approximation (SA) method for solving stochastic convex composite optimization (SCCO) problems has remained an open challenge. The difficulty arises from the fact that the stochastic multi-cut model, constructed as the pointwise maximum of individual stochastic linearizations, provides a biased estimate of the objective function, with the error being uncontrollable. This paper introduces multi-cut SA methods for solving SCCO problems, achieving near-optimal convergence rates. The cutting-plane models used in these methods are the pointwise maxima of appropriately chosen one-cut models. To the best of our knowledge, these are the first multi-cut SA methods specifically designed for SCCO problems.
Talk 2: Data-driven Policies for Two-Stage Stochastic Linear Programs
Speaker: Chhavi Sharma
Abstract: A stochastic program typically involves several parameters including deterministic first-stage parameter, and stochastic second-stage elements that serve as input data. These programs are usually re-solved whenever there is a change in any of the input parameters. For example, a stochastic dispatch problem is solved multiple times a day due to fluctuations in electricity prices, demand, and renewable energy availability. However, in practical situations, quick decision-making is crucial, and solving a stochastic program from scratch for every change in input data can be computationally costly. This work addresses this challenge for two-stage stochastic linear programs (2-SLPs) with varying first-stage right-hand sides. We employ data-driven approaches to first construct a novel piecewise linear difference of max-affine policy (PLDC) for deterministic linear programs. This is achieved by leveraging optimal basis matrices from previous solutions and the piecewise linear nature of the optimal solution trajectory. This PLDC policy retains optimal solutions for previously encountered parameters and is expected to provide good-quality solutions for new parameters. Our developed policy applies directly to the extensive form of 2-SLP. When algorithms such as the L-shaped method are applied to solve 2-SLP, we construct the policy using local outer approximations of the recourse function and optimal basis matrices from previous solves. We assess the performance of our policy through both numerical and theoretical analyses. Our numerical experiments show small feasibility and relative optimal gap at solutions returned by the policy.
Talk 3: Statistical Robustness Analysis of In-CVaR Based Regression
Speaker: Yulei You
Abstract: Based on the interval conditional value-at-risk (In-CVaR) proposed in Liu & Pang (2022), this paper investigates the robustness of In-CVaR based regression under data contamination and perturbation. To quantify robustness under data contamination, we introduce the concept of “distributional breakdown point (BP) value”. Our results provide upper and lower bounds for the distributional BP value, which can be widely applied to classic regression tasks, including linear regression, piecewise affine regression, and feedforward neural network regression with homogeneous activation functions. Furthermore, we demonstrate that under data perturbation, the In-CVaR based estimator is qualitatively robust against optimization if and only if the largest portion of the loss is trimmed. Overall, this research complements the robust analysis of In-CVaR and shows that In-CVaR outperforms conditional value-at-risk and sample average approximation in terms of robustness for regression tasks. This talk is based on a joint work with Prof. Junyi Liu at Tsinghua University.
