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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.
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Nonsmooth Projection-Free Optimization with Functional Constraints
This paper presents a subgradient-based algorithm for constrained nonsmooth convex optimization that does not require projections onto the feasible set. While the well-established Frank-Wolfe algorithm and its variants already avoid projections, they are primarily designed for smooth objective functions. In con- trast, our proposed algorithm can handle nonsmooth problems with general convex functional inequality constraints. It achieves an ϵ-suboptimal solution in O(ϵ^−2) iterations, with each iteration requiring only a single (potentially inexact) Linear Minimization Oracle (LMO) call and a (possibly inexact) subgra- dient computation. This performance is consistent with existing lower bounds. Similar performance is observed when deterministic subgradients are replaced with stochastic subgradients. In the special case where there are no functional inequality constraints, our algorithm competes favorably with a recent nonsmooth projection-free method designed for constraint-free problems. Our approach uti- lizes a simple separation scheme in conjunction with a new Lagrange multiplier update rule.
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- Award ID(s):
- 1824418
- PAR ID:
- 10494716
- Editor(s):
- arXiv:2311.11180v1
- Publisher / Repository:
- arXiv:2311.11180v1
- Date Published:
- Subject(s) / Keyword(s):
- Projection-free optimization, Frank-Wolfe method, Nonsmooth convex optimization, Stochastic optimization, Functional constraints
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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