Penalty-Based Smoothing of Convex Nonsmooth Supremum Functions with Accelerated Inertial Dynamics

We propose a penalty-based smoothing framework for convex nonsmooth functions with a supremum structure. The regularization yields a differentiable surrogate with controlled approximation error, a single-valued dual maximizer, and explicit gradient formulas. We then study an accelerated inertial dynamic with vanishing damping driven by a time-dependent regularized function whose parameter decreases to zero. Under mild integrability and boundedness conditions on the regularization schedule, we establish an accelerated $\mathcal{O}(t^{-2})$ decay estimate for the regularized residual and, in the regime $\alpha >3$, a sharper $o(t^{-2})$ decay together with weak convergence of trajectories to a minimizer of the original nonsmooth problem via an Opial-type argument. Applications to multiobjective optimization (through Chebyshev/max scalarization) and to distributionally robust optimization (via entropic regularization over ambiguity sets) illustrate the scope of the framework.