Distribution-Induced Conic Geometry and Distributionally Conic Robust Optimization: A Unified Primal–Dual Framework

This paper proposes a novel optimization framework—Distributionally Conic Robust Optimization (DCRO). Unlike classical cone programming, which treats the geometric structure as a deterministic set, and unlike traditional distributed blob optimization, which only introduces distributed uncertainty at the objective or constraint function level, this paper, for the first time, embeds the "distributed structure" into the "cone structure" itself. By constructing distributed induced cones, distributed polar cones, and desired geometric operators, a structural coupling mechanism between probabilistic uncertainty and convex geometry is established. This paper proves that distributed induced cones maintain closed convexity and possess a stable representation theorem under appropriate compactness and continuity conditions; further, it establishes a strong duality theorem and a polarization equivalence theorem for distributed cones; and it provides the collapse condition for distributed cones, revealing its essential connection with classical cone programming. This theory unifies cone programming, distributed blob optimization, and polarized geometry, providing a new fundamental framework for uncertain optimization.