Learning Optimization Proxies for Sequential Contextual Stochastic Programs: An Order Fulfillment Application

Sequential contextual stochastic programs model real-time decision systems in which each time epoch commits to an action under uncertainty whose consequences propagate into future decisions. In many practical contexts, these programs require obtaining solutions rapidly as new information becomes available. These problems can be represented through scenario approximations to be solved by off-the-shelf optimization solvers, which achieve high decision quality offline but typically run in seconds to minutes per instance, falling short of the sub-second responses that peak periods of planning require. This paper develops a learning-based optimization proxy: a scenario-embedded neural network trained offline on solver-generated labels, paired online with a decoder that enforces feasibility, replacing the per-epoch solve with a single forward pass. The framework is specialized to omnichannel order fulfillment, where each arriving order requires a sub-second assignment of products to distribution centers and carrier services under stochastic delivery times and future demand. A two-stage contextual stochastic program is introduced to formulate this problem, and its contextual sample average approximation (C-SAA) supplies the offline labels, while a composite training loss combines label imitation, a constraint-violation penalty, and self-supervised cost alignment. In a calibrated simulator built from JD.com transactional records, a detailed computational study is provided. The proxy reduces decision latency by roughly 2800x relative to the online finite-sample C-SAA reference and improves over it by 3.3% in realized fulfillment cost. Relative to established fulfillment policies, the proxy lowers total realized cost by at least 10.7% and roughly halves the late-delivery rate.

The Proxy Benders Decomposition

Benders decomposition is a fundamental framework for solving large-scale mixed-integer optimization problems with complicating variables that, when fixed, yield significantly easier subproblems. However, classical Benders decomposition repeatedly solves highly similar subproblems and often exhibits zigzagging behavior across iterations, leading to slow convergence in large-scale settings. Motivated by the repetitive structure and parametric nature of Benders subproblems, this paper introduces the proxy Benders decomposition (Proxy-BD), a new decomposition framework in which subproblem optimization is replaced by certified optimization proxies rather than repeated exact solves. The proposed proxy follows a self-supervised predict-project-and-complete mechanism that produces dual-feasible solutions for generating provably valid Benders cuts. The framework preserves the theoretical validity of the decomposition independently of prediction quality through a projection-and-completion certification layer. A formal characterization of proxy-induced cuts is established, and the framework naturally extends to modern decomposition schemes, including branch-and-Benders-cut algorithms. Computational experiments on large-scale facility location and network design problems demonstrate that Proxy-BD substantially reduces the computational effort of subproblems while maintaining near-optimal solution quality. On large-scale uncapacitated facility location instances up to 2000x2000, Proxy-BD achieves median optimality gaps below 0.5%, yields up to 161x median speedups, and reduces the number of generated cuts by more than 240x on the largest instances. The computational gains consistently increase with recourse complexity, indicating that proxy-based inference scales substantially more favorably than repeated exact subproblem optimization in large-scale decomposition settings.

Are Stochastic Multi-objective Bandits Harder than Single-objective Bandits?

Multi-objective bandits have attracted increasing attention because of their broad applicability and mathematical elegance, where the reward of each arm is a multi-dimensional vector rather than a scalar. This naturally introduces Pareto order relations and Pareto regret. A long-standing question in this area is whether performance is fundamentally harder to optimize because of this added complexity. A recent surprising result shows that, in the adversarial setting, Pareto regret is no larger than classical regret; however, in the stochastic setting, where the regret notion is different, the picture remains unclear. In fact, existing work suggests that Pareto regret in the stochastic case increases with the dimensionality. This controversial yet subtle phenomenon motivates our central question: \emph{are multi-objective bandits actually harder than single-objective ones?} We answer this question in full by showing that, in the stochastic setting, Pareto regret is in fact governed by the maximum sub-optimality gap \(g^\dagger\), and hence by the minimum marginal regret of order \(Ω(\frac{K\log T}{g^\dagger})\). We further develop a new algorithm that achieves Pareto regret of order \(O(\frac{K\log T}{g^\dagger})\), and is therefore optimal. The algorithm leverages a nested two-layer uncertainty quantification over both arms and objectives through upper and lower confidence bound estimators. It combines a top-two racing strategy for arm selection with an uncertainty-greedy rule for dimension selection. Together, these components balance exploration and exploitation across the two layers. We also conduct comprehensive numerical experiments to validate the proposed algorithm, showing the desired regret guarantee and significant gains over benchmark methods.