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Recent efforts to accelerate first-order methods have focused on convexoptimization problems that satisfy a geometric property known as error-boundcondition, which covers a broad class of problems, including piece-wise linearprograms and strongly convex programs. Parameter-free first-order methods thatemploy projection-free updates have the potential to broaden the benefit ofacceleration. Such a method has been developed for unconstrained convexoptimization but is lacking for general constrained convex optimization. Wepropose a parameter-free level-set method for the latter constrained case basedon projection-free subgradient decent that exhibits accelerated convergence forproblems that satisfy an error-bound condition. Our method maintains a separatecopy of the level-set sub-problem for each level parameter value and restartsthe computation of these copies based on objective function progress. Applyingsuch a restarting scheme in a level-set context is novel and results in analgorithm that dynamically adapts the precision of each copy. This property iskey to extending prior restarting methods based on static precision that havebeen proposed for unconstrained convex optimization to handle constraints. Wereport promising numerical performance relative to benchmark methods.

A Parameter-free and Projection-free Restarting Level Set Method for Adaptive Constrained Convex Optimization Under the Error Bound Condition