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We consider the problem of minimizing a convex function over the intersectionof finitely many simple sets which are easy to project onto. This is animportant problem arising in various domains such as machine learning. The maindifficulty lies in finding the projection of a point in the intersection ofmany sets. Existing approaches yield an infeasible point with aniteration-complexity of $O(1/\varepsilon^2)$ for nonsmooth problems with noguarantees on the in-feasibility. By reformulating the problem through exactpenalty functions, we derive first-order algorithms which not only guaranteesthat the distance to the intersection is small but also improve the complexityto $O(1/\varepsilon)$ and $O(1/\sqrt{\varepsilon})$ for smooth functions. Forcomposite and smooth problems, this is achieved through a saddle-pointreformulation where the proximal operators required by the primal-dualalgorithms can be computed in closed form. We illustrate the benefits of ourapproach on a graph transduction problem and on graph matching.

Convex optimization over intersection of simple sets: improved convergence rate guarantees via an exact penalty approach