Uncertain Cone-Operator Embedded Programming: A Unified Cone-Order Polarization Framework for Robust Conic Duality

This paper proposes a novel optimization framework called uncertain cone operator embedded programming. Unlike traditional robust optimization, which treats uncertainty as an external perturbation of a linear operator, this study embeds uncertainty within a cone dual structure. By introducing two new concepts—"operator-valued cone mapping" and "cone-ordered polarization operator"—a unified first-order optimization theory is established. Under this theory, general cone programming and robust linear programming become two boundary degenerate cases. This paper establishes an existence theorem, a cone-ordered polarization theorem, a strong duality theorem, and a cone-uncertainty equivalence convexity theorem, proving that this framework is geometrically equivalent to a new polar-cone overlapping structure. This result provides a new foundation for the geometric stability and structural robustness of uncertain systems.