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# Efficient First-order Methods for Convex Optimization with Strongly Convex Function Constraints

Dec 2022

Convex function constrained optimization has received growing researchinterests lately. For a special convex problem which has strongly convexfunction constraints, we develop a new accelerated primal-dual first-ordermethod that obtains an $\Ocal(1/\sqrt{\vep})$ complexity bound, improving the$\Ocal(1/{\vep})$ result for the state-of-the-art first-order methods. The keyingredient to our development is some novel techniques to progressivelyestimate the strong convexity of the Lagrangian function, which enablesadaptive step-size selection and faster convergence performance. In addition,we show that the complexity is further improvable in terms of the dependence onsome problem parameter, via a restart scheme that calls the accelerated methodrepeatedly. As an application, we consider sparsity-inducing constrainedoptimization which has a separable convex objective and a strongly convex lossconstraint. In addition to achieving fast convergence, we show that therestarted method can effectively identify the sparsity pattern (active-set) ofthe optimal solution in finite steps. To the best of our knowledge, this is thefirst active-set identification result for sparsity-inducing constrainedoptimization.

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