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We propose a new randomized algorithm for solving convex optimizationproblems that have a large number of constraints (with high probability).Existing methods like interior-point or Newton-type algorithms are hard toapply to such problems because they have expensive computation and storagerequirements for Hessians and matrix inversions. Our algorithm is based onnonlinear rescaling (NLR), which is a primal-dual-type algorithm by Griva andPolyak {[{Math. Program., 106(2):237-259, 2006}]}. NLR introduces an equivalentproblem through a transformation of the constraint functions, minimizes thecorresponding augmented Lagrangian for given dual variables, and then uses thisminimizer to update the dual variables for the next iteration. The primalupdate at each iteration is the solution of an unconstrained finite summinimization problem where the terms are weighted by the current dualvariables. We use randomized first-order algorithms to do these primal updates,for which they are especially well suited. In particular, we use the scaleddual variables as the sampling distribution for each primal update, and we showthat this distribution is the optimal one among all probability distributions.We conclude by demonstrating the favorable numerical performance of ouralgorithm.

A Randomized Nonlinear Rescaling Method in Large-Scale Constrained Convex Optimization