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We consider a convex optimization problem with many linear inequalityconstraints. To deal with a large number of constraints, we provide a penaltyreformulation of the problem, where the penalty is a variant of the one-sidedHuber loss function with two penalty parameters. We study the infeasibilityproperties of the solutions of penalized problems for nonconvex and convexobjective functions, as the penalty parameters vary with time. Then, we proposea random incremental penalty method for solving the original problem, andinvestigate its convergence properties for convex and strongly convex objectivefunctions. We show that the iterates of the method converge to a solution ofthe original problem almost surely and in expectation for suitable choices ofthe penalty parameters and the stepsize. Also, we establish convergence rate ofthe method in terms of the expected function values by utilizing appropriatelydefined weighted averages of the iterates. We show $O(\ln^{1/2+\epsilon}k/{\sqrt k})$-convergence rate when the objective function is convex and$O(\ln^{\epsilon} k/k)$-convergence rate when the objective function isstrongly convex, with $\epsilon>0$ being an arbitrarily small scalar. } To thebest of our knowledge, these are the first results on the convergence rate forthe penalty-based incremental subgradient method with time-varying penaltyparameters.

Huber Loss-Based Penalty Approach to Problems with Linear Constraints