Recent efforts to accelerate first-order methods have focused on convex optimization problems that satisfy a geometric property known as error-bound condition, which covers a broad class of problems, including piece-wise linear programs and strongly convex programs. Parameter-free first-order methods that employ projection-free updates have the potential to broaden the benefit of acceleration. Such a method has been developed for unconstrained convex optimization but is lacking for general constrained convex optimization. We propose a parameter-free level-set method for the latter constrained case based on projection-free subgradient method that exhibits accelerated convergence for problems that satisfy an error-bound condition. Our method maintains a separate copy of the level-set sub-problem for each level parameter value and restarts the computation of these copies based on objective function progress. Applying such a restarting scheme in a level-set context is novel and results in an algorithm that dynamically adapts the precision of each copy. This property is key to extending prior restarting methods based on static precision that have been proposed for unconstrained convex optimization to handle constraints. We report promising numerical performance relative to benchmark methods.

Stochastic convex optimization problems with expectation constraints (SOECs) are encountered in statistics and machine learning, business, and engineering. The SOEC objective and constraints contain expectations defined with respect to complex distributions or large data sets, leading to high computational complexity when solved by the algorithms that use exact functions and their gradients. Recent stochastic first order methods exhibit low computational complexity when handling SOECs but guarantee near-feasibility and near-optimality only at convergence. These methods may thus return highly infeasible solutions when heuristically terminated, as is often the case, due to theoretical convergence criteria being highly conservative. This issue limits the use of first order methods in several applications where the SOEC constraints encode implementation requirements. We design a stochastic feasible level set method (SFLS) for SOECs that has low complexity and emphasizes feasibility before convergence. Specifically, our level-set method solves a root-finding problem by calling a novel first order oracle that computes a stochastic upper bound on the level-set function by extending mirror descent and online validation techniques. We establish that SFLS maintains a high-probability feasible solution at each root-finding iteration and exhibits favorable complexity compared to state-of-the-art deterministic feasible level set and stochastic subgradient methods. Numerical experiments on three diverse applications highlight how SFLS finds feasible solutions with small optimality gaps with lower complexity than the former approaches.

Product brands employ shopper marketing (SM) strategies to convert shoppers along the path to purchase. Traditional marketing mix models (MMMs), which leverage regression techniques and historical data, can be used to predict the component of sales lift due to SM tactics. The resulting predictive model is a critical input to plan future SM strategies. The implementation of traditional MMMs, however, requires significant ad-hoc manual intervention due to their limited flexibility in (i) explicitly capturing the temporal link between decisions; (ii) accounting for the interaction between business rules and past (sales and decision) data during the attribution of lift to SM; and (iii) ensuring that future decisions adhere to business rules. These issues necessitate MMMs with tailored structures for specific products and retailers, each requiring significant hand-engineering to achieve satisfactory performance -- a major implementation challenge. We propose an SM Optimization and Inverse Learning Engine (SMOILE) that combines optimization and inverse reinforcement learning to streamline implementation. SMOILE learns a model of lift by viewing SM tactic choice as a sequential process, leverages inverse reinforcement learning to explicitly couple sales and decision data, and employs an optimization approach to handle a wide-array of business rules. Using a unique dataset containing sales and SM spend information across retailers and products, we illustrate how SMOILE standardizes the use of data to prescribe future SM decisions. We also track an industry benchmark to showcase the importance of encoding SM lift and decision structures to mitigate spurious results when uncovering the impact of SM decisions.