Penalty-Based Smoothing of Convex Nonsmooth Supremum Functions with Accelerated Inertial Dynamics
Samir Adly, Juan José Maulén, Emilio Vilches
TL;DR
The paper develops a penalty-based smoothing framework for convex nonsmooth supremum functions and analyzes a nonautonomous inertial dynamic driven by the time-varying regularized objective. By constructing $\varphi_\mu$ with a dual-regularization and an explicit envelope gradient, it achieves a controlled approximation error and enables accelerated continuous-time convergence results via $\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+\nabla\varphi_{\mu(t)}(x)=0$. Under mild decay assumptions on the smoothing schedule $\mu(t)$, the authors prove an $O(t^{-2})$ decay of the regularized residual and, for $\alpha>3$, a stronger $o(t^{-2})$ rate with weak convergence to a minimizer of the original problem (via Opial). The framework applies to Chebyshev scalarization in multiobjective optimization and to distributionally robust optimization with entropic regularization, illustrating broad applicability to robust and scalarized optimization models. Overall, the study provides a principled route to combining smoothing, acceleration, and asymptotic consistency in nonsmooth convex optimization problems, with potential implications for scalable algorithms in robust settings.
Abstract
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 $α>3$, 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.