🤖 AI Summary
This work addresses a fundamental bottleneck in stochastic combinatorial optimization (SCO): the difficulty of handling complex constraints expressed as expectations—such as empirical conditional value-at-risk (CVaR)—within standard SCO frameworks. We are the first to unify single-level expectation constraints and bilevel composite constraints into a cohesive SCO formulation, enabling risk-averse optimization and higher-order moment portfolio selection in data-driven settings. We propose the first primal-dual algorithm for this class of problems that simultaneously offers theoretical guarantees and practical efficacy. Our method integrates stochastic gradient estimation, momentum updates, dual coupling, nested sampling, and variance reduction, achieving the optimal $O(1/sqrt{N})$ convergence rate under both single-level and bilevel constraints. It naturally supports joint modeling and optimization of multiple expectation-based constraints. This work establishes a new benchmark for expectation-constrained SCO, substantially enhancing the feasibility, accuracy, and applicability of risk-sensitive optimization.
📝 Abstract
Stochastic compositional optimization (SCO) has attracted considerable attention because of its broad applicability to important real-world problems. However, existing works on SCO assume that the projection within a solution update is simple, which fails to hold for problem instances where the constraints are in the form of expectations, such as empirical conditional value-at-risk constraints. We study a novel model that incorporates single-level expected value and two-level compositional constraints into the current SCO framework. Our model can be applied widely to data-driven optimization and risk management, including risk-averse optimization and high-moment portfolio selection, and can handle multiple constraints. We further propose a class of primal-dual algorithms that generates sequences converging to the optimal solution at the rate of O ( 1 √ N ) under both single-level expected value and two-level compositional constraints, where N is the iteration counter, establishing the benchmarks in expected value constrained SCO.
