Projection-Free Methods for Stochastic Simple Bilevel Optimization with Convex Lower-level Problem

In this paper, we study a class of stochastic bilevel optimization problems, also known as stochastic simple bilevel optimization, where we minimize a smooth stochastic objective function over the optimal solution set of another stochastic convex optimization problem. We introduce novel stochastic bilevel optimization methods that locally approximate the solution set of the lower-level problem via a stochastic cutting plane, and then run a conditional gradient update with variance reduction techniques to control the error induced by using stochastic gradients. For the case that the upper-level function is convex, our method requires $\tilde{\mathcal{O}}(\max\{1/\epsilon_f^{2},1/\epsilon_g^{2}\})$ stochastic oracle queries to obtain a solution that is $\epsilon_f$-optimal for the upper-level and $\epsilon_g$-optimal for the lower-level. This guarantee improves the previous best-known complexity of $\mathcal{O}(\max\{1/\epsilon_f^{4},1/\epsilon_g^{4}\})$. Moreover, for the case that the upper-level function is non-convex, our method requires at most $\tilde{\mathcal{O}}(\max\{1/\epsilon_f^{3},1/\epsilon_g^{3}\})$ stochastic oracle queries to find an $(\epsilon_f, \epsilon_g)$-stationary point. In the finite-sum setting, we show that the number of stochastic oracle calls required by our method are $\tilde{\mathcal{O}}(\sqrt{n}/\epsilon)$ and $\tilde{\mathcal{O}}(\sqrt{n}/\epsilon^{2})$ for the convex and non-convex settings, respectively, where $\epsilon=\min \{\epsilon_f,\epsilon_g\}$

Pricing and Coordination Strategy for Green Supply Chain Under Two Production Modes

The purpose of this study is to address the issue of coordination and pricing in a single-period and three-stage green supply chain in which green products and non-green products exist together and can be substituted with each other in the market. We examine the equilibrium results for two production modes, green production mode and hybrid production mode, in the cooperative and non-cooperative game to demonstrate the importance of entering supply chain members in collaboration. Theoretical analysis shows that different production costs lead the manufacturer to decide on different production modes when customers have further evaluations about various types of products. Furthermore, the results indicate that the system's performance in a cooperative game is better than that in a non-cooperative game, implying that supply chain members will respond positively to collaboration as their profit is higher than that under the non-cooperative strategy. The cooperative pricing strategy implemented by the Rubinstein bargaining model can provide the Pareto optimal solution for the supply chain system's profit and members' profits considering different production modes. Finally, the proposed model is applied to a generated numerical example to validate the suggested coordinated pricing strategy's validity and results.Immediate accessThis item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]