A Unified Measure–Cone Coupled Variational Framework for Total Variation Distributionally Robust Conic Optimization

This paper establishes a novel optimization theoretical framework—measure-cone coupled variational robust optimization theory. This theory simultaneously characterizes the convex cone geometry and the probabilistic distribution perturbation structure within a unified variational system, thereby achieving an endogenous coupling between structural feasibility and distribution uncertainty. Unlike traditional cone programming, which only deals with deterministic parameters, and unlike existing Total Variation Partial Brullough Optimization, which only perturbs the objective function, this paper proposes constructing a two-layer coupled variational operator between the vector space and the measure space, and establishing its unified extremum condition. The core results show that, under the total variation uncertainty radius rho, the primal problem is equivalent to performing a thickening transformation of the support function on the dual cone, and the robust term geometrically manifests as the measure expansion of the dual cone boundary. This paper proves strong duality, stability, robust value continuity, and error bounds, and presents a unified equation in variational inequality form. This theory not only unifies cone programming and partial Brullough Optimization but also reveals the deep duality between probabilistic perturbations and convex cone geometry, providing a new fundamental theoretical framework for high-dimensional uncertain optimization.