Dual-Inertial Bregman Methods with Energy-Based Safeguarding

Abstract

We develop dual-inertial Bregman proximal methods for convex optimization and analyze their convergence through an exact signed-drift representation. For unrestarted iterations, one-sided directional curvature control yields a last-iterate $\cO((1+\log k)/k^2)$ rate, improving to $\cO(k^{-2})$ under weighted summability. A computable energy-based safeguard guarantees a global $\cO(N^{-1})$ objective-gap bound, counting all proximal oracle calls. Under a stepsize condition and local quadratic growth with suitable kernel regularity, the safeguarded methods restart finitely and eventually converge linearly. Numerical experiments illustrate the methods on entropy regression and nonnegative least squares.