Search for: All records

Augmented Lagrangian (AL) methods have proven remarkably useful in solving optimization problems with complicated constraints. The last decade has seen the development of overall complexity guarantees for inexact AL variants. Yet, a crucial gap persists in addressing nonsmooth convex constraints. To this end, we present a smoothed augmented Lagrangian (AL) framework where nonsmooth terms are progressively smoothed with a smoothing parameter $$\eta_k$$. The resulting AL subproblems are $$\eta_k$$-smooth, allowing for leveraging accelerated schemes. By a careful selection of the inexactness level  (for inexact subproblem resolution), the penalty parameter $$\rho_k$$, and smoothing parameter $$\eta_k$$ at epoch k, we derive rate and complexity guarantees of  $$\tilde{\mathcal{O}}(1/\epsilon^{3/2})$$ and $$\tilde{\mathcal{O}}(1/\epsilon)$$  in convex and strongly convex regimes for computing an -optimal solution, when $$\rho_k$$ increases at a geometric rate, a significant improvement over the best available guarantees for AL schemes for convex programs with nonsmooth constraints. Analogous guarantees are developed for settings with $$\rho_k=\rho$$ as well as $$\eta_k=\eta$$. Preliminary numerics on a fused Lasso problem display promise. 

We consider a continuous-valued simulation optimization (SO) problem, where a simulator is built to optimize an expected performance measure of a real-world system while parameters of the simulator are estimated from streaming data collected periodically from the system. At each period, a new batch of data is combined with the cumulative data and the parameters are re-estimated with higher precision. The system requires the decision variable to be selected in all periods. Therefore, it is sensible for the decision-maker to update the decision variable at each period by solving a more precise SO problem with the updated parameter estimate to reduce the performance loss with respect to the target system. We define this decision-making process as the multi-period SO problem and introduce a multi-period stochastic approximation (SA) framework that generates a sequence of solutions. Two algorithms are proposed: Re-start SA (ReSA) reinitializes the stepsize sequence in each period, whereas Warm-start SA (WaSA) carefully tunes the stepsizes, taking both fewer and shorter gradient-descent steps in later periods as parameter estimates become increasingly more precise. We show that under suitable strong convexity and regularity conditions,ReSAandWaSAachieve the best possible convergence rate in expected sub-optimality either when an unbiased or a simultaneous perturbation gradient estimator is employed, whileWaSAaccrues significantly lower computational cost as the number of periods increases. In addition, we present theregularizedReSA, which obviates the need to know the strong convexity constant and achieves the same convergence rate at the expense of additional computation. 

With a manifold growth in the scale and intricacy of systems, the challenges of parametric misspecification become pronounced. These concerns are further exacerbated in compositional settings, which emerge in problems complicated by modeling risk and robustness. In “Data-Driven Compositional Optimization in Misspecified Regimes,” the authors consider the resolution of compositional stochastic optimization problems, plagued by parametric misspecification. In considering settings where such misspecification may be resolved via a parallel learning process, the authors develop schemes that can contend with diverse forms of risk, dynamics, and nonconvexity. They provide asymptotic and rate guarantees for unaccelerated and accelerated schemes for convex, strongly convex, and nonconvex problems in a two-level regime with extensions to the multilevel setting. Surprisingly, the nonasymptotic rate guarantees show no degradation from the rate statements obtained in a correctly specified regime and the schemes achieve optimal (or near-optimal) sample complexities for general T-level strongly convex and nonconvex compositional problems.